Open and Closed Sets in topological space

1. Formula
   Observation

   The symbol τ is written in the definition of open set and in the example topology.

2. Audio
   Observation

   The speaker says, "If tau is a topology..."

$\tau$

A topology on a set X.

A collection of subsets of X.

1. Formula
   Observation

   The symbol X is used to denote the underlying set in the topological space (X, $\tau$) and in the example.

2. Audio
   Observation

   The speaker says, "Consider a topological space X and tau..."

X

The underlying set of a topological space.

A set.

1. Formula
   Observation

   The symbol y is used to denote a subset of X in the general definition and in the specific example.

2. Audio
   Observation

   The speaker says, "and let Y be a subset of X..."

1. The speaker uses uppercase 'Y' verbally while writing lowercase 'y' on the board.

y

A subset of the underlying set X.

A subset of X.

1. Formula
   Observation

   The symbol $\subseteq$ is used to indicate that y is a subset of X.

$\subseteq$

Subset or equal to.

Set inclusion relation.

1. Formula
   Observation

   The phrase "is a member of" is used to indicate set membership, equivalent to the symbol $\in$.

$\in$

Is an element of / is a member of.

Set membership relation.

1. Formula
   Observation

   The notation $y^c$ is used to denote the complement of the set y.

$y^c$

The complement of the set y with respect to the universal set X.

A subset of X.

1. Formula
   Observation

   The board shows $X = {a, b, c, d, e}$.

2. Audio
   Observation

   The speaker refers to the entire set X when computing complements and checking whether X is open or closed.

X

The underlying set of the topological space, explicitly given as {a, b, c, d, e}.

1. Formula
   Observation

   The board shows τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}.

2. Audio
   Observation

   The speaker repeatedly says that a set is or is not a member of tau.

τ

The topology on X, i.e. the collection of open sets used to decide openness and closedness.

1. Formula
   Observation

   The board shows $y = {a, b, c}$ and y ⊆ X.

2. Audio
   Observation

   The speaker checks whether y is open and then whether y is closed.

y

A subset of X under test, explicitly $y = {a, b, c}$.

1. Formula
   Observation

   The board writes $y^c = X - y = {a,b,c,d,e} - {a,b,c} = {d,e}$.

2. Audio
   Observation

   The speaker says, 'We simply look for y complement' and computes it as the elements of X minus the elements of y.

$y^c$

The complement of y relative to X, computed as X - y.

1. Formula
   Observation

   The board lists ∅ as an element of τ and later writes $X^c = X - X$ = ∅.

2. Audio
   Observation

   The speaker calls ∅ the empty set and uses it as another example of a clopen set.

∅

The empty set; here it is both an element of the topology and the complement of X in X.

1. Formula
   Observation

   The board writes $U = {c,d}$ and then U ⊆ X and $U \in τ$.

2. Audio
   Observation

   The speaker introduces 'another subset of X, say U equals to C D' and concludes that U is open.

U

Another subset of X under test, explicitly $U = {c,d}$.

1. Formula
   Observation

   Written text: "If $\tau$ is a topology, then the members of $\tau$ are called the open sets of the topology $\tau$"

2. Audio
   Observation

   Speaker reads the definition of an open set.

In a topological space $(X, \tau)$, the elements (members) of the topology $\tau$ are defined as the open sets.

1. $(X, \tau)$ is a topological space.

2. $y$ is a subset of $X$.

3. $y$ is a member of the topology $\tau$.

1. Formula
   Observation

   Written text: "If a given set y is open, then $y^c$ is said to be closed."

2. Audio
   Observation

   Speaker reads the definition of a closed set.

A set is defined as closed if its complement is an open set.

1. $y$ is a subset of $X$.

2. $y$ is an open set in the topological space $(X, \tau)$.

1. Definition of Open Set

1. Formula
   Observation

   Written text: "OPEN and CLOSED — CLOPEN SET" and "In any set X, the set itself and the empty set is called a clopen set."

2. Audio
   Observation

   Speaker explains that a set both open and closed is called a clopen set, and gives the whole set and empty set as examples.

A set that is simultaneously open and closed is called a clopen set. In any topological space, the entire set $X$ and the empty set $\emptyset$ are always clopen.

1. $y$ is a subset of $X$.

2. $y$ is an open set.

3. $y$ is a closed set.

1. Definition of Open Set
2. Definition of Closed Set

1. Formula
   Observation

   The board uses statements such as y ⊆ X, y ∉ τ ⇒ y is not open, and X ⊆ X, $X \in τ$ ⇒ X is open.

2. Audio
   Observation

   The speaker decides openness by checking whether the subset belongs to τ.

In this video, a subset of X is called open exactly when it is a member of the displayed topology τ. The examples show the test directly: $y = {a,b,c}$ is not open because y ∉ τ, while X and $U = {c,d}$ are open because they belong to τ.

1. The ambient set is $X = {a,b,c,d,e}$.

2. The topology is τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}.

1. Formula
   Observation

   For y, the board computes $y^c = {d,e}$, notes {d,e} ∉ τ, and concludes y is not closed.

2. Formula
   Observation

   For X, the board computes $X^c$ = ∅, notes ∅ ∈ τ, and concludes X is closed.

3. Audio
   Observation

   The speaker says, 'Now let's also see if it is closed or not. We simply look for y complement' and later applies the same method to X.

The video defines closedness by looking at the complement relative to X: a set is closed when its complement is open, i.e. when the complement lies in τ. This is applied twice: y is not closed because $y^c = {d,e}$ ∉ τ, and X is closed because $X^c$ = ∅ ∈ τ.

1. Complements are taken relative to the fixed universe X.

2. Openness of the complement is judged using membership in τ.

1. Criterion for an open set in this worked example

1. Formula
   Observation

   The board writes $y^c = X - y = {a,b,c,d,e} - {a,b,c} = {d,e}$.

2. Formula
   Observation

   The board writes $X^c = X - X$ = ∅.

3. Audio
   Observation

   The speaker explains the complement as 'the entire set X minus the subset y' and later as 'X minus X'.

The video uses complement notation $A^c$ to mean the set difference X - A inside the fixed ambient set X. It demonstrates the computation by removing the listed elements of A from the listed elements of X.

1. A must be a subset of the chosen universe X.

2. Here $X = {a,b,c,d,e}$.

1. Formula
   Observation

   After proving X is open and X is closed, the board writes CLOPEN SET.

2. Audio
   Observation

   The speaker says, 'Now this means that the set X is a clopen set' and adds that ∅ is also clopen.

A clopen set is a set that is both open and closed in the given topology. The video first labels X as clopen after showing $X \in τ$ and $X^c$ = ∅ ∈ τ, then states that ∅ is also clopen because ∅ ∈ τ and its complement is $X \in τ$.

1. Both openness and closedness are judged in the same topological space (X, τ).

1. Criterion for an open set in this worked example
2. Criterion for a closed set via complement

1. Formula
   Observation

   The board states "$U \subseteq X$ and $U \in T$ => U is open".

In the displayed setup, a subset U of X is called open when U belongs to the topology T. The video presents this as the working test for openness in the example.

1. U must be a subset of X.

2. Membership in T is the decisive condition for openness in this example.

1. Formula
   Observation

   The hand writes "$U^c = X - U$".

2. Audio
   Observation

   The speaker explains the complement as the elements of X minus the elements of U.

The complement of U is obtained by removing from X all elements that belong to U. In this clip it is used as the intermediate object whose membership in T determines whether U is closed.

1. The complement is taken relative to the ambient set X.

1. Formula
   Observation

   After computing \{a,b,e\} $\notin T$, the board writes "=> U is not closed".

2. Audio
   Observation

   The speaker says, "Now let's see if it is also closed or not. We simply look for U complement ... And this implies that U is not closed."

1. The general theorem "U is closed iff $U^c$ is open" is not written out in full form in this clip; what is directly shown is the applied test through the complement.

To decide whether U is closed, the video computes $U^c$ and checks whether that complement belongs to the topology T. If the complement is not in T, the example concludes that U is not closed.

1. The method assumes openness is being read off membership in T.

2. The complement is computed relative to X.

1. Criterion for a set to be open
2. Complement of a subset relative to X

1. Formula
   Observation

   The heading "CLOPEN SET" appears above $\emptyset$.

1. The clip does not define the term "clopen" in words during this segment; it only labels $\emptyset$ with that term.

The board uses the label "CLOPEN SET" for $\emptyset$, indicating that the empty set is being treated as both open and closed in the surrounding lesson.

1. This is a label shown on screen rather than a fully restated definition within the clip.

1. Formula
   Observation

   The board shows $y = {a,b,c}$, y ⊆ X, y ∉ τ, and the conclusion '=> y is not open.'

2. Audio
   Observation

   The spoken setup leads into checking whether y is closed after already establishing non-openness.

For $y = {a,b,c}$ in the displayed space, y is not open.

1. $X = {a,b,c,d,e}$.

2. τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}.

3. $y = {a,b,c}$ ⊆ X.

4. y ∉ τ.

A specific instance claim about the named set y in the given topological space.

1. Formula
   Observation

   The board computes $y^c = {d,e}$, writes {d,e} ∉ τ, and concludes '=> y is not closed'.

2. Audio
   Observation

   The speaker says, 'we can see that the set D E is not a member of tau. This implies that Y is not closed.'

For $y = {a,b,c}$, the complement $y^c = {d,e}$ is not in τ, so y is not closed.

1. Closedness is tested by whether the complement lies in τ.

2. $y^c = X - y = {d,e}$.

3. {d,e} ∉ τ.

A specific instance claim about the named set y.

1. Audio
   Observation

   The speaker says, 'Therefore, Y is neither open nor closed.'

2. Formula
   Observation

   This follows from the two preceding written conclusions on the board: y is not open and y is not closed.

The set $y = {a,b,c}$ is neither open nor closed in (X, τ).

1. y is not open because y ∉ τ.

2. y is not closed because $y^c = {d,e}$ ∉ τ.

A conjunction of two specific instance claims about y.

1. Formula
   Observation

   The board writes 'Every set is a subset of itself, X ⊆ X', then 'and $X \in τ$', then '=> X is open'.

2. Audio
   Observation

   The speaker says, 'Recall that every set is a subset of itself... we can see that X is also a member of tau. This implies that X is an open set...'

The whole space X is open in the topology τ.

1. X ⊆ X.

2. $X \in τ$.

A general fact instantiated to the specific space shown: the ambient set X itself.

1. Formula
   Observation

   The board writes $X^c = X - X$ = ∅ and ∅ ∈ τ, then concludes '=> X is closed'.

2. Audio
   Observation

   The speaker says, 'the empty set is a member of tau. So this also shows that X is closed.'

The whole space X is closed because its complement is ∅ and ∅ ∈ τ.

1. $X^c = X - X$ = ∅.

2. ∅ ∈ τ.

3. Closedness is defined by having complement in τ.

A specific instance claim about the ambient set X.

1. Formula
   Observation

   After the two conclusions '=> X is open' and '=> X is closed', the board writes CLOPEN SET.

2. Audio
   Observation

   The speaker says, 'Now this means that the set X is a clopen set.'

The set X is clopen in (X, τ).

1. X is open.

2. X is closed.

A specific instance claim about the ambient set X.

1. Formula
   Observation

   The board writes ∅ under the heading CLOPEN SET.

2. Audio
   Observation

   The speaker says, 'Another example is the empty set. The empty set is also a clopen set because it is a member of tau and its complement is the set X which is a member of tau.'

The empty set ∅ is clopen in (X, τ).

1. ∅ ∈ τ.

2. ∅^$c = X$.

3. $X \in τ$.

A specific instance claim about the empty set in the given space.

1. Formula
   Observation

   The board writes $U = {c,d}$, U ⊆ X, $U \in τ$, and '=> U is open'.

2. Audio
   Observation

   The speaker says, 'We can see that U is a subset of X and we can also see that U is a member of tau. So this implies that U is open.'

For $U = {c,d}$, U is open in (X, τ).

1. $U = {c,d}$ ⊆ X.

2. $U \in τ$.

A specific instance claim about the named set U.

1. Formula
   Observation

   The board already contains "$U \subseteq X$ and $U \in T$ => U is open" while U = \{c,d\} is displayed.

1. The explicit membership check "\{c,d\} $\in T$" is not rewritten in this clip; the openness statement is carried over from earlier content.

For the subset U = \{c,d\}, the video treats U as open because it satisfies the displayed criterion $U \subseteq X$ and $U \in T$.

1. U = \{c,d\}.

2. $U \subseteq X$.

3. $U \in T$.

Specific example about the displayed set U.

1. Formula
   Observation

   The completed line reads "$U^c = X - U$ = \{a,b,c,d,e\} - \{c,d\} = \{a,b,e\} $\notin T$" followed by "=> U is not closed".

2. Audio
   Observation

   The speaker states that the complement is not a member of tau and therefore U is not closed.

The subset U = \{c,d\} is not closed in the given topology because its complement \{a,b,e\} is not in T.

1. U = \{c,d\}.

2. X = \{a,b,c,d,e\}.

3. $U^c$ = \{a,b,e\}.

4. \{a,b,e\} $\notin T$.

Specific example about the displayed set U.

1. Formula
   Observation

   Written steps: "$y = {a, b, c}$", "$y \subseteq X$", "$y \notin \tau$", "=> y is not open."

2. Audio
   Observation

   Speaker walks through the example, noting that {a, b, c} is a subset of X but not in the topology tau, concluding it is not open.

1. Expression

   $$X = \{a, b, c, d, e\}, \tau = \{\emptyset, \{a\}, \{c,d\}, \{a,c,d\}, \{b,c,d,e\}, X\}$$
   Explanation

   State the given topological space and its topology.

   Justification

   Given in the example.

   Shown in the video
2. Expression

   $$y = \{a, b, c\}$$
   Explanation

   Define a specific subset y of X.

   Justification

   Chosen for the example.

   Shown in the video
3. Expression

   $$y \subseteq X$$
   Explanation

   Verify that y is a subset of X.

   Justification

   All elements of y are in X.

   Shown in the video
4. Expression

   $$y \notin \tau$$
   Explanation

   Check if y is a member of the topology tau.

   Justification

   The set {a, b, c} is not listed in the elements of tau.

   Shown in the video
5. Expression

   $$\Rightarrow y \text{ is not open}$$
   Explanation

   Conclude that y is not an open set.

   Justification

   By the definition of an open set, a subset must be a member of the topology to be open.

   Shown in the video

The set $y = {a, b, c}$ is not an open set in the given topological space.

1. Formula
   Observation

   The board sequentially writes $y^c = X - y$, substitutes {a,b,c,d,e} - {a,b,c}, obtains {d,e}, then writes {d,e} ∉ τ and '=> y is not closed'.

2. Audio
   Observation

   The speaker narrates each step: find y complement, subtract elements of y from X, get de, observe de is not in tau, conclude y is not closed.

1. Expression

   $$y^c = X - y$$
   Explanation

   Start from the definition of complement relative to the ambient set X.

   Justification

   Definition of relative complement used in the video.

   Shown in the video
2. Expression

   $$X - y = \{a,b,c,d,e\} - \{a,b,c\}$$
   Explanation

   Substitute the explicit sets shown on the board.

   Justification

   Direct substitution from $X = {a,b,c,d,e}$ and $y = {a,b,c}$.

   Shown in the video
3. Expression

   $$= \{d,e\}$$
   Explanation

   Remove the elements a, b, c from X, leaving d and e.

   Justification

   Set difference computation.

   Shown in the video
4. Expression

   $$\{d,e\} \notin \tau$$
   Explanation

   Compare the result with the listed members of the topology.

   Justification

   Inspection of τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}.

   Shown in the video
5. Expression

   $$\Rightarrow y \text{ is not closed}$$
   Explanation

   Since the complement is not open, y fails the closed-set test.

   Justification

   Closedness criterion: A is closed iff $A^c \in τ$.

   Shown in the video

$y = {a,b,c}$ is not closed in the given topological space.

1. Formula
   Observation

   The board first writes X ⊆ X and $X \in τ$, concluding X is open; then writes $X^c = X - X$ = ∅ and ∅ ∈ τ, concluding X is closed.

2. Audio
   Observation

   The speaker explicitly performs both checks in order and then says X is a clopen set.

1. Expression

   $$X \subseteq X$$
   Explanation

   Note that the ambient set is a subset of itself.

   Justification

   Basic subset fact stated on the board: every set is a subset of itself.

   Shown in the video
2. Expression

   $$X \in \tau$$
   Explanation

   Check the displayed topology and see that X is listed as a member.

   Justification

   Inspection of τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}.

   Shown in the video
3. Expression

   $$\Rightarrow X \text{ is open}$$
   Explanation

   Because X is a subset of X and belongs to τ, it satisfies the openness test used in the video.

   Justification

   Open-set criterion A open iff $A \in τ$.

   Shown in the video
4. Expression

   $$X^c = X - X = \emptyset$$
   Explanation

   Compute the complement of X relative to itself.

   Justification

   Definition of relative complement and set difference.

   Shown in the video
5. Expression

   $$\emptyset \in \tau$$
   Explanation

   Check the topology again and see that the empty set is listed as a member.

   Justification

   Inspection of τ.

   Shown in the video
6. Expression

   $$\Rightarrow X \text{ is closed}$$
   Explanation

   Since the complement of X is open, X is closed.

   Justification

   Closed-set criterion A closed iff $A^c \in τ$.

   Shown in the video

X is both open and closed, hence clopen.

1. Audio
   Observation

   The speaker states that ∅ is clopen because ∅ ∈ τ and its complement is $X \in τ$.

2. Formula
   Observation

   The board writes ∅ under CLOPEN SET, while earlier lines already established ∅ ∈ τ and $X \in τ$.

1. The equality ∅^$c = X$ is spoken rather than separately written out on the board.

1. Expression

   $$\emptyset \in \tau$$
   Explanation

   The empty set is one of the listed members of the topology.

   Justification

   Inspection of τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}.

   Shown in the video
2. Expression

   $$\emptyset^c = X$$
   Explanation

   The complement of the empty set relative to X is the whole space.

   Justification

   Standard relative-complement identity; the speaker explicitly says its complement is the set X.

   Derived from the video
3. Expression

   $$X \in \tau$$
   Explanation

   The whole space is also listed as a member of the topology.

   Justification

   Already shown earlier in the video.

   Shown in the video
4. Expression

   $$\Rightarrow \emptyset \text{ is clopen}$$
   Explanation

   Both ∅ and its complement are open, so ∅ is both open and closed.

   Justification

   Definition of clopen set.

   Derived from the video

∅ is a clopen set in (X, τ).

1. Audio
   Observation

   The speaker says, "Now let's see if it is also closed or not. We simply look for U complement."

2. Formula
   Observation

   The board successively writes "$U^c = X - U$", then substitutes the sets, then concludes "=> U is not closed".

1. Expression

   $$U^c = X - U$$
   Explanation

   Introduce the complement of U relative to X as the object to inspect for closedness.

   Justification

   Definition of complement used in the video.

   Shown in the video
2. Expression

   $$X - U = \{a,b,c,d,e\} - \{c,d\}$$
   Explanation

   Substitute the explicit ambient set and the explicit subset into the complement formula.

   Justification

   Direct substitution from the displayed sets.

   Shown in the video
3. Expression

   $$\{a,b,c,d,e\} - \{c,d\} = \{a,b,e\}$$
   Explanation

   Remove c and d from X, leaving a, b, and e.

   Justification

   Set difference computation.

   Shown in the video
4. Expression

   $$\{a,b,e\} \notin T$$
   Explanation

   Check the resulting complement against the topology T and find that it is not a member.

   Justification

   Explicit written non-membership statement on the board.

   Shown in the video
5. Expression

   $$U \text{ is not closed}$$
   Explanation

   Conclude that U fails to be closed because its complement is not open in the sense used by the example.

   Justification

   Applied closedness test via complement membership in T.

   Shown in the video

The example proves that U = \{c,d\} is not closed by showing $U^c$ = \{a,b,e\} $\notin T$.

1. Formula
   Observation

   Written text: "EXAMPLE", "$X = {a, b, c, d, e}$", "$\tau = {\emptyset, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}$"

2. Audio
   Observation

   Speaker introduces the example and states the set X and topology tau.

Identify whether a specific subset is an open set in a given topological space.

1. X = \{a, b, c, d, e\}

2. $\tau$ = \{$\emptyset$, \{a\}, \{c,d\}, \{a,c,d\}, \{b,c,d,e\}, X\}

3. y = \{a, b, c\}

Determine if the subset $y = {a, b, c}$ is an open set.

1. Expression

   $$y \subseteq X$$
   Explanation

   Check if y is a subset of X.

   Justification

   All elements of y are present in X.

   Shown in the video
2. Expression

   $$y \notin \tau$$
   Explanation

   Check if y is a member of the topology tau.

   Justification

   The set {a, b, c} is not one of the elements listed in tau.

   Shown in the video
3. Expression

   $$\Rightarrow y \text{ is not open}$$
   Explanation

   Apply the definition of an open set.

   Justification

   Since y is not a member of tau, it cannot be an open set.

   Shown in the video

The set $y = {a, b, c}$ is not an open set.

The conclusion directly follows from the definition of open sets in a topological space.

1. Formula
   Observation

   The board displays X, τ, $y = {a,b,c}$, y ⊆ X, y ∉ τ, then $y^c = {d,e}$, {d,e} ∉ τ, and the conclusions that y is not open and not closed.

2. Audio
   Observation

   The speaker walks through both the openness check and the complement-based closedness check for y.

Given $X = {a,b,c,d,e}$ and τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}, determine whether $y = {a,b,c}$ is open and whether it is closed.

1. $X = {a,b,c,d,e}$.

2. τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}.

3. $y = {a,b,c}$ ⊆ X.

Decide whether y is open and whether y is closed in (X, τ).

1. Expression

   $$y \notin \tau$$
   Explanation

   Compare y with the listed members of the topology.

   Justification

   Direct inspection of τ.

   Shown in the video
2. Expression

   $$\Rightarrow y \text{ is not open}$$
   Explanation

   Since openness is membership in τ, y fails to be open.

   Justification

   Open-set criterion used in the video.

   Shown in the video
3. Expression

   $$y^c = X - y = \{a,b,c,d,e\} - \{a,b,c\} = \{d,e\}$$
   Explanation

   Compute the complement of y relative to X.

   Justification

   Definition of relative complement and explicit set subtraction.

   Shown in the video
4. Expression

   $$\{d,e\} \notin \tau$$
   Explanation

   Check whether the complement belongs to the topology.

   Justification

   Inspection of τ.

   Shown in the video
5. Expression

   $$\Rightarrow y \text{ is not closed}$$
   Explanation

   Because the complement is not open, y is not closed.

   Justification

   Closed-set criterion A closed iff $A^c \in τ$.

   Shown in the video

y is neither open nor closed.

The conclusion matches the two written implications on the board: '=> y is not open.' and '=> y is not closed'.

1. Formula
   Observation

   The board writes X ⊆ X, $X \in τ$, => X is open; then $X^c = X - X$ = ∅, ∅ ∈ τ, => X is closed; then CLOPEN SET and ∅.

2. Audio
   Observation

   The speaker asks 'What about the set X?', proves X is open and closed, then says X is clopen and gives ∅ as another clopen example.

Using the same X and τ, determine whether X is open, whether X is closed, and identify clopen examples mentioned in the video.

1. $X = {a,b,c,d,e}$.

2. τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}.

3. Every set is a subset of itself.

Classify X with respect to openness and closedness, and note the clopen examples stated by the speaker.

1. Expression

   $$X \subseteq X$$
   Explanation

   The ambient set is trivially a subset of itself.

   Justification

   General subset fact stated on the board.

   Shown in the video
2. Expression

   $$X \in \tau$$
   Explanation

   X appears explicitly among the members of the topology.

   Justification

   Inspection of τ.

   Shown in the video
3. Expression

   $$\Rightarrow X \text{ is open}$$
   Explanation

   Membership in τ gives openness.

   Justification

   Open-set criterion.

   Shown in the video
4. Expression

   $$X^c = X - X = \emptyset$$
   Explanation

   Compute the complement of X in itself.

   Justification

   Relative complement definition.

   Shown in the video
5. Expression

   $$\emptyset \in \tau$$
   Explanation

   The empty set is listed in the topology.

   Justification

   Inspection of τ.

   Shown in the video
6. Expression

   $$\Rightarrow X \text{ is closed}$$
   Explanation

   Since $X^c$ is open, X is closed.

   Justification

   Closed-set criterion.

   Shown in the video
7. Expression

   $$X \text{ is clopen}$$
   Explanation

   X is both open and closed.

   Justification

   Definition of clopen set.

   Shown in the video
8. Expression

   $$\emptyset \text{ is also clopen}$$
   Explanation

   The speaker adds ∅ as another example because ∅ ∈ τ and ∅^$c = X \in τ$.

   Justification

   Same openness and complement tests applied to ∅.

   Shown in the video

X is open, X is closed, so X is clopen; ∅ is also stated to be clopen.

The board explicitly writes '=> X is open', '=> X is closed', and 'CLOPEN SET', followed by ∅.

1. Formula
   Observation

   The board writes $U = {c,d}$, U ⊆ X, $U \in τ$, and '=> U is open'.

2. Audio
   Observation

   The speaker says, 'Let's play around with another subset of X, say U equals to C D... this implies that U is open.'

1. The clip ends before any closedness check for U is shown.

For $U = {c,d}$ in the same topological space, determine whether U is open.

1. $X = {a,b,c,d,e}$.

2. τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}.

3. $U = {c,d}$.

Decide whether U is open in (X, τ).

1. Expression

   $$U \subseteq X$$
   Explanation

   Check that U is indeed a subset of the ambient set.

   Justification

   Direct comparison of elements.

   Shown in the video
2. Expression

   $$U \in \tau$$
   Explanation

   Look for {c,d} among the listed members of the topology.

   Justification

   Inspection of τ.

   Shown in the video
3. Expression

   $$\Rightarrow U \text{ is open}$$
   Explanation

   Membership in τ implies openness.

   Justification

   Open-set criterion used throughout the video.

   Shown in the video

$U = {c,d}$ is open.

The final written line on the board is '=> U is open'.

1. Formula
   Observation

   The board shows U = \{c,d\}, the openness criterion, the complement computation, and the final conclusion "=> U is not closed".

2. Audio
   Observation

   The narration walks through checking whether U is also closed and ends by inviting viewers to try other subsets of X.

1. The full topology T is not listed explicitly inside this clip, so only the membership facts actually shown can be recorded.

Given the displayed subset U = \{c,d\} of X, determine whether U is open and whether it is closed.

1. U = \{c,d\}.

2. X = \{a,b,c,d,e\} (visible in the complement computation).

3. The openness rule shown is $U \subseteq X$ and $U \in T$ => U is open.

4. The computed complement is \{a,b,e\}.

5. \{a,b,e\} $\notin T$.

Test whether U is open and whether U is closed in the given topological setting.

1. Expression

   $$U \subseteq X \text{ and } U \in T \Rightarrow U \text{ is open}$$
   Explanation

   Use the displayed criterion to treat U as open.

   Justification

   Already-written statement on the board at the start of the clip.

   Shown in the video
2. Expression

   $$U^c = X - U$$
   Explanation

   Switch from openness to closedness by examining the complement.

   Justification

   Method stated verbally and written on the board.

   Shown in the video
3. Expression

   $$U^c = \{a,b,c,d,e\} - \{c,d\} = \{a,b,e\}$$
   Explanation

   Compute the complement explicitly.

   Justification

   Set difference using the displayed elements of X and U.

   Shown in the video
4. Expression

   $$\{a,b,e\} \notin T$$
   Explanation

   Compare the complement with the topology T.

   Justification

   Explicit non-membership written on the board.

   Shown in the video
5. Expression

   $$U \text{ is not closed}$$
   Explanation

   Conclude failure of closedness from the complement test.

   Justification

   Final implication written on the board.

   Shown in the video

U = \{c,d\} is treated as open, but it is not closed.

The verification shown is the complement calculation: since $U^c$ = \{a,b,e\} and \{a,b,e\} $\notin T$, the example concludes U is not closed.

1. Animation
   Observation

   A hand writes 'STUDYAFRIC', 'Topology 1.3', and 'OPEN AND CLOSED SETS' sequentially on a white background.

1. Hand

2. Marker

3. Text

1. Text appears letter by letter as if being written by the hand.

1. White background remains constant.

Introduces the channel name, course number, and topic of the video.

1. Animation
   Observation

   A hand writes the definitions of open, closed, and clopen sets on the screen.

1. Hand

2. Marker

3. Text

1. Definitions appear sequentially on the screen.

1. White background remains constant.

Visually presents the formal mathematical definitions being discussed by the speaker.

1. Animation
   Observation

   A hand writes the example sets X, tau, and y, followed by the logical deduction steps.

1. Hand

2. Marker

3. Text

1. Example components and deduction steps appear sequentially.

1. White background remains constant.

Visually structures the example problem and its step-by-step solution.

1. Diagram
   Observation

   At the start, the whiteboard shows $y = {a, b, c}$, y ⊆ X, y ∉ τ, and the blue conclusion '=> y is not open.'

1. $y = {a,b,c}$

2. y ⊆ X

3. y ∉ τ

4. blue implication text

1. The board already presents the non-membership statement y ∉ τ.

2. The conclusion '=> y is not open.' is visible in blue.

1. The ambient universe and topology are not yet fully visible in the opening frame.

2. The logical status of y as a subset of X remains fixed.

The opening visual encodes the definition-based test for openness: because y is not listed in τ, the video marks y as not open.

1. Animation
   Observation

   A hand writes $y^c = X - y$, then substitutes {a,b,c,d,e} - {a,b,c}, and finishes with {d,e}.

2. Audio
   Observation

   The narration tracks the writing step by step while explaining complement as X minus y.

1. Hand and marker

2. Expression $y^c = X - y$

3. Explicit sets {a,b,c,d,e} and {a,b,c}

4. Result {d,e}

1. The expression is built left to right from definition to substitution to simplified result.

2. The intermediate subtraction form appears before the final set {d,e}.

1. The complement is always taken relative to the same universe X.

2. The set y remains {a,b,c} throughout the computation.

The animation makes the abstract definition $A^c = X - A$ concrete by literally removing the elements of y from X and leaving exactly the elements d and e.

1. Diagram
   Observation

   The camera reveals the full board including $X = {a,b,c,d,e}$, τ = {∅, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}, the prior y-lines, and the new line {d,e} ∉ τ with red '=> y is not closed'.

1. X

2. τ

3. y

4. $y^c = {d,e}$

5. {d,e} ∉ τ

6. red conclusion text

1. The previously hidden definitions of X and τ become visible.

2. The board now connects the computed complement {d,e} to the topology list.

3. A red implication states that y is not closed.

1. The earlier blue conclusion that y is not open remains on the board.

2. The computed complement {d,e} does not change once written.

This reveal supplies the missing reference list needed to judge membership in τ, turning the complement computation into a closedness decision.

1. Animation
   Observation

   The hand writes 'Every set is a subset of itself, X ⊆ X', then 'and $X \in τ$', then blue '=> X is open'; afterwards it writes $X^c = X - X$ = ∅, 'and ∅ ∈ τ', then red '=> X is closed'.

1. Sentence 'Every set is a subset of itself'

2. X ⊆ X

3. $X \in τ$

4. Blue '=> X is open'

5. $X^c = X - X$ = ∅

6. ∅ ∈ τ

7. Red '=> X is closed'

1. The argument first establishes openness of X from direct membership in τ.

2. It then switches to the complement test and establishes closedness of X.

3. Color coding separates the open conclusion (blue) from the closed conclusion (red).

1. The same topology τ is used for both conclusions.

2. The universe X remains fixed while its self-complement is computed.

The visual sequence demonstrates that the whole space passes both tests: it is listed as open directly, and its complement is the empty set, which is also listed as open, so X is closed too.

1. Diagram
   Observation

   Green text 'CLOPEN SET' is written beneath the two conclusions about X, and then ∅ is written below it.

2. Audio
   Observation

   The speaker says X is a clopen set and adds that the empty set is another example.

1. Green heading CLOPEN SET

2. Symbol ∅

1. A new green label summarizes the previous two results about X.

2. The symbol ∅ is added as a second example under the same heading.

1. The preceding proofs that X is open and X is closed remain visible above.

2. The topology τ is unchanged.

The board groups together sets that are both open and closed, first instantiating the concept with X and then with ∅.

1. Animation
   Observation

   The hand writes $U = {c,d}$, then U ⊆ X and $U \in τ$, followed by blue '=> U is open'.

1. $U = {c,d}$

2. U ⊆ X

3. $U \in τ$

4. Blue conclusion '=> U is open'

1. A new subset U is introduced.

2. The board checks subset status and topology membership for U.

3. A blue openness conclusion is written.

1. The same X and τ from earlier remain the reference structures.

2. No complement computation for U is shown within this clip.

The last visual event reuses the same openness criterion as before: once U is seen as an element of τ, the video immediately concludes that U is open.

1. Diagram
   Observation

   The opening frame shows the green heading "CLOPEN SET", the symbol $\emptyset$, the red note "=> X is closed", the black line "U = \{c,d\}", and the blue conclusion "=> U is open".

1. "CLOPEN SET"

2. $\emptyset$

3. "=> X is closed"

4. U = \{c,d\}

5. "$U \subseteq X$ and $U \in T$ => U is open"

1. No new writing occurs yet; the board presents previously established conclusions.

1. The displayed subset remains U = \{c,d\}.

2. The openness conclusion for U is already present on screen.

The visual layout separates earlier results (clopen example and openness of U) from the upcoming closedness check.

1. Animation
   Observation

   A hand writes the complement line step by step: first $U^c$, then X - U, then the explicit set subtraction, then \{a,b,e\}, then $\notin T$, and finally the red conclusion "=> U is not closed".

1. Handwritten formula line

2. Set symbols

3. Red implication text

1. The expression grows from "$U^c$" to the full chain "$U^c = X - U$ = \{a,b,c,d,e\} - \{c,d\} = \{a,b,e\} $\notin T$".

2. A new red conclusion is added beneath the formula.

1. The ambient set X and the subset U remain fixed throughout the computation.

2. The topology symbol T remains the reference family for membership checks.

The animation makes the logical order explicit: define complement, compute it, test membership in T, then infer non-closedness.

1. Animation
   Observation

   The mathematical board disappears and promotional graphics appear: a red "SUBSCRIBE NOW" banner, a bell icon, a YouTube-style like/share graphic, and the text "Thanks for watching" plus "Visit JW.ORG to get answers to life big questions".

2. Audio
   Observation

   The speaker asks viewers to subscribe, click the bell icon, comment, like, share, and thanks them for watching.

1. Subscribe banner

2. Bell icon

3. Like/share graphic

4. Closing text

1. The content shifts from topology notation to channel promotion and closing remarks.

1. No further mathematical statements are introduced after the closedness conclusion.

This interval contains no additional mathematical development beyond the example already completed.

1. Formula
   Observation

   The board does not test y directly against τ for closedness; instead it computes $y^c$ and then checks {d,e} ∉ τ.

2. Audio
   Observation

   The speaker says, 'Now let's also see if it is closed or not. We simply look for y complement'.

One might think a set is closed exactly when the set itself belongs to the topology.

The video uses the complementary criterion: A is closed when $A^c$ is open, i.e. when $A^c \in τ$. For y, the relevant object checked against τ is {d,e}, not y itself.

1. Audio
   Observation

   The speaker concludes, 'Therefore, Y is neither open nor closed.'

2. Formula
   Observation

   The board separately shows '=> y is not open.' and '=> y is not closed'.

One might assume every subset of a topological space must be either open or closed.

This example shows a subset $y = {a,b,c}$ that fails both tests: y ∉ τ and $y^c$ ∉ τ, so it is neither open nor closed.

1. Formula
   Observation

   The board writes CLOPEN SET only after proving both '=> X is open' and '=> X is closed'.

2. Audio
   Observation

   The speaker says X is a clopen set and then gives ∅ as another example because both it and its complement are in τ.

One might read 'clopen' as meaning either open or closed, or as a third unrelated category.

In the video, clopen is introduced as the conjunction of the two previously proved properties: a set is clopen when it is both open and closed in the same topology.

1. Audio
   Observation

   After already having U labeled open, the speaker separately asks whether it is "also closed or not" and then performs a different test using the complement.

2. Formula
   Observation

   The board keeps the openness statement and the closedness conclusion as separate lines.

One might think that once a set is shown to be open, it is automatically closed as well.

The example explicitly treats openness and closedness as separate properties: U is shown as open, yet the complement test leads to the conclusion that U is not closed.

1. Formula
   Observation

   The definition of a closed set explicitly relies on the concept of an open set ("If a given set y is open...").

The definition of a closed set depends on the prior definition of an open set.

1. Formula
   Observation

   The term 'CLOPEN' is formed by combining 'CLOSED' and 'OPEN', and the definition states a set must be both.

The definition of a clopen set requires a set to be open.

1. Formula
   Observation

   The term 'CLOPEN' is formed by combining 'CLOSED' and 'OPEN', and the definition states a set must be both.

The definition of a clopen set requires a set to be closed.

1. Audio
   Observation

   Speaker says, "So let's take some examples to understand the concept of open and closed sets."

2. Formula
   Observation

   The example applies the condition $y \in \tau$ to determine if a set is open.

The example demonstrates how to apply the definition of an open set to a specific topological space.

1. Formula
   Observation

   Closedness is decided by checking whether $y^c$ or $X^c$ belongs to τ, the same topology used for openness.

2. Audio
   Observation

   The speaker first discusses openness and then says closedness is checked by looking for the complement.

The closed-set test depends on the open-set test, because 'closed' is defined here as 'having an open complement,' and openness is membership in τ.

1. Formula
   Observation

   The derivations for y and X both begin by computing $A^c = X - A$ before checking membership in τ.

To apply the complement-based closedness criterion, one must first know how to form the relative complement $A^c = X - A$.

1. Formula
   Observation

   The heading CLOPEN SET appears only after the two separate conclusions that X is open and X is closed.

2. Audio
   Observation

   The speaker says X is clopen precisely because both properties have just been shown.