The Math Sorcerer · YouTube · 6:19
This video segment provides the formal definition of a topology on a non-empty set X. It explains that a topology is a collection of 'open' subsets satisfying three key properties: the whole set and empty set are open, arbitrary unions of open sets are open, and finite intersections of open sets are open. The concept is built step-by-step on a digital whiteboard. This 150-second whiteboard lecture introduces the axiomatic definition of a topology on a nonempty set X and then defines a topological space as the pair (X, τ). The board states three requirements: X and ∅ are open; arbitrary unions of open sets are open; and finite intersections of open sets are open. It then works a concrete finite example with and τ={∅,{a},{a,b},X}, checking each axiom in turn and concluding that τ is indeed a topology. Near the end, a second example on the same set is started but not completed within the clip. This introductory topology whiteboard clip defines a topological space as a pair (X, τ), where τ is a collection of subsets of X called open subsets. It lists three required properties: ∅ and X must be in τ, arbitrary unions of members of τ must remain in τ, and finite intersections of members of τ must remain in τ. Two finite examples on are compared. The collection {∅, {a}, {a, b}, X} is treated as satisfying the conditions, while {∅, {a, b}, {a, c}, X} fails because {a, b} ∩ {a, c} = {a} is not in τ. The speaker concludes that τ is not a topology on X and restates the three axioms.
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Generated from the video's visuals and explanation; not verbatim speech.
We begin by introducing the core concept: the definition of a topology on a given set.
Let be a non-empty set. This set serves as the foundational space upon which our topological structure will be built.
A topology on is formally defined as a specific collection of subsets of . These subsets are designated as "open" sets. The quotation marks emphasize that 'open' is a defined term within this context, not necessarily relying on a metric or distance function.
For a collection of subsets to qualify as a topology, it must satisfy three fundamental properties. The first property requires that both the entire set itself and the empty set are included in the collection of open sets.
The second property concerns unions. It states that the union of *any* family of open sets—whether finite or infinite—must also result in an open set belonging to the topology.
The third and final property deals with intersections. Unlike unions, this property is restricted to *finite* intersections. The intersection of any finite number of open sets must also be an open set within the topology.
To summarize, if we have a non-empty set and we can identify a collection of its subsets (denoted by ) that satisfies these three conditions, then is a valid topology on , and its elements are the open sets.
The clip opens on a digital blackboard with the general setup already written: for a nonempty set , a topology on is a collection of subsets of called open subsets, and this collection must satisfy three listed conditions.
The lecturer then names the resulting structure. A topological space is the ordered pair , where is the underlying set and is the chosen collection of open subsets.
To make the definition concrete, the video moves to a small finite example. The set is , and the proposed topology is .
The question now is whether this particular really satisfies the axioms of a topology on .
First, axiom (1) is checked directly: both and appear in the list defining , so they are open by definition.
Next, axiom (2) concerns unions. The speaker tests whether taking unions of members of this finite collection can produce a set outside the collection. Representative cases are discussed, including unions involving itself, and the conclusion is that every such union remains one of the listed members of .
Then axiom (3) is checked with a finite intersection. The board computes , and since , this intersection is still open. The lecturer notes that the other finite intersections behave the same way.
Because all three axioms hold, the example collection is confirmed to be a topology on , so is a topological space.
At the very end, the lecturer begins a second example using the same underlying set and starts to introduce a new , but the clip stops before that second collection is fully written or analyzed.
The board presents the general setup: a topological space is written , and is described as the collection of all open subsets of . Three conditions are listed at the upper right: are open, closure under unions, and closure under finite intersections.
A second example is introduced on the same underlying set . The proposed collection is . The speaker first checks the easy requirements: and are both present, so the first condition holds.
Next, the union condition is tested informally. The speaker observes that , and since is already in , this displayed union does not create a problem. The spoken reasoning suggests condition two is acceptable for the example under discussion.
The decisive test is the finite-intersection condition. The speaker intersects the two nontrivial proposed open sets and computes , because is the only element they have in common.
The result is then checked against the listed collection . It is not one of , so the board records . This violates closure under finite intersections.
Therefore the conclusion is written and spoken: is not a topology on . A single failed finite intersection is enough to disprove the claim, even though the earlier empty-set, whole-set, and union checks did not fail.
The clip closes by restating the definition in words. A topology on is a collection of subsets called open subsets that satisfies three properties: and must be open, the union of any number of open sets must remain in , and any finite intersection of open sets must also remain in .
A topology on a non-empty set X is a collection of subsets of X, called 'open' sets, that satisfies three specific axioms regarding the inclusion of X and the empty set, closure under arbitrary unions, and closure under finite intersections.
The entire set X and the empty set must both be members of the topology . This ensures the space itself and the void are considered 'open'.
If you take any collection of open sets (finite or infinite) from the topology, their union must also be an open set in the topology.
If you take a finite number of open sets from the topology, their intersection must also be an open set in the topology. Note that this only applies to finite intersections.
The Greek letter tau () is commonly used to denote the topology itself, which is the collection of all open subsets satisfying the three axioms.
A topology on a nonempty set X is a chosen collection τ of subsets of X, called open subsets, that satisfies three axioms: X and ∅ belong to τ; arbitrary unions of members of τ are again in τ; and finite intersections of members of τ are again in τ.
The video defines a topological space not as the bare set X but as the ordered pair (X, τ), where τ is a topology on X. This emphasizes that the same set can carry different topologies.
In this lesson, a subset of X is open exactly when it is a member of the chosen topology τ. Openness is therefore defined axiomatically by membership in τ, not by distance or intervals.
For , the collection τ={∅,{a},{a,b},X} is tested against the three topology axioms. Both ∅ and X are in τ; unions of members stay in τ; and the sample finite intersection {a}∩{a,b}={a} is again in τ. Hence τ is a topology on X.
The axioms distinguish the two operations: closure is required for arbitrary unions of open sets, but only for finite intersections. The example reinforces this by checking a finite intersection explicitly.
Near the end, the lecturer starts another example on the same set and begins to define a new τ, but the clip ends before that second topology is completed or verified.
A topological space is written as , where is the underlying set and is a collection of subsets of called open subsets. In this clip, the board states and then tests candidate collections against the topology axioms.
The clip lists three required properties for to be a topology on : (1) and are in ; (2) unions of any number of members of remain in ; (3) finite intersections of members of remain in . The final spoken summary repeats exactly these three conditions.
On , the collection is checked on the board. It contains and , and the displayed finite intersection remains in . The example is presented as satisfying the three topology conditions.
For the same underlying set , the collection passes the first check because and are included, and the displayed union stays in . However, the finite intersection produces a set not listed in , so is not a topology on .
A topology must satisfy all three axioms simultaneously. The second example shows that passing the empty-set/whole-set condition and a union check does not guarantee that is a topology. The single violation after computing is sufficient to conclude that is not a topology on .
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
Written as 'Let '.
Spoken as 'let X be a non-empty set'.
X
A non-empty set on which a topology is defined.
Set
Written as '' in '' and later as an element of the topology.
The empty set.
Set
Written as ' = \{ \}'.
Spoken as 'the actual topology, let's use the letter tau'.
The collection of all open subsets that satisfies the three properties of a topology.
Collection of sets
The board shows the underlined symbol X in statements such as "Let X ≠ ∅" and later in the example "".
The speaker refers to it verbally as "capital X" and "our set".
X
A nonempty underlying set on which a topology is defined; in the worked examples it is instantiated as .
A set; in the examples, a finite set with three elements.
The axiom line reads "(1) X, ∅ are open." and the example topology contains ∅ as an element.
The speaker says "the empty set" while listing members of τ.
∅
The empty set, required to be one of the open sets in any topology.
A set.
The board writes τ = {all open subsets of X} and later τ = {∅, {a}, {a,b}, X}.
The speaker says "tau, which is the collection of what we say are open subsets" and then lists the elements of τ in the example.
τ
A topology on X, i.e. a collection of subsets of X designated as open and satisfying the three axioms.
A subset of the power set of X; in the example, a four-element collection of subsets of {a,b,c}.
The board writes "topological space (X, τ)".
The speaker says "a topological space is actually just a pair" and identifies the two components as the set and τ.
(X, τ)
The ordered pair consisting of a set X and a topology τ on X; this pair is called a topological space.
An ordered pair whose first component is a set and second component is a topology on that set.
The example line reads "".
The speaker says "So this will be the elements A, B, C."
a,b,c
The three listed elements of the example set X.
Elements of the finite set .
The example topology includes {a} as one of its members.
The speaker says "the singleton set A" while writing the list for τ.
{a}
The singleton subset of X containing only a; in the example it is declared open because it belongs to τ.
A subset of X.
The example topology includes {a,b} as one of its members.
The speaker says "the set containing A and B" while listing τ.
{a,b}
The two-element subset of X containing a and b; in the example it is declared open because it belongs to τ.
A subset of X.
The worked check writes "{a} ∩ {a,b} = {a} ∈ τ".
The speaker says "if you take the intersection ... A intersect AB" while verifying condition (3).
∩
Set intersection used to test closure under finite intersections of open sets.
An operation on subsets of X.
The speaker repeatedly says "If we take the union of any number of these sets" and gives examples such as taking X union A.
The axiom line states "(2) the union of any family of open sets is open."
No explicit union symbol is written in the extracted frames; the operation is conveyed by the axiom text and spoken examples.
union
Set union, used in the second topology axiom requiring closure under arbitrary unions of open sets.
An operation on families of subsets of X.
Speaker defines a topology as a collection of 'open' subsets satisfying three properties.
Written definition and three numbered conditions on the board.
A topology on a non-empty set X is a collection of subsets of X, called 'open' subsets, that satisfy three specific properties regarding the whole set, empty set, unions, and finite intersections.
X is a non-empty set
is a collection of subsets of X
Written as '(1) X, are open.'
Spoken as 'both the entire set X and the empty set as well are open'.
The entire set X and the empty set must both be included in the collection of open sets.
Written as '(2) the union of any family of open sets is open.'
Spoken as 'the union of any family of open sets in X is also open in X'.
The union of any collection (finite or infinite) of open sets must also be an open set.
Written as '(3) the finite intersection of any collection of open sets is open.'
Spoken as 'the finite intersection of any collection of open sets is also open'.
The intersection of any finite collection of open sets must also be an open set.
The board states: "Let X ≠ ∅. A topology on X is a collection of 'open subsets' of X which satisfy the following." followed by three numbered conditions.
The speaker explains that τ is "the collection of what we say are open subsets" and introduces the term topological space as the pair (X, τ).
For a nonempty set X, a topology on X is a chosen collection τ of subsets of X, called the open subsets, that satisfies three axioms: X and ∅ belong to τ; arbitrary unions of members of τ remain in τ; and finite intersections of members of τ remain in τ. The video presents this as the foundational definition before giving examples.
X is nonempty.
τ is a collection of subsets of X.
The three listed axioms must all hold.
The board writes "topological space (X, τ)".
The speaker says "a topological space is actually just a pair" and identifies the pair as the set together with τ.
A topological space is not just the underlying set X but the ordered pair (X, τ), where τ is a topology on X. The video emphasizes that the word "topological space" usually refers to this pair structure.
X is a set.
τ is a topology on X.
The definition line calls τ a collection of "open subsets" of X, and the example checks label members of τ as open.
The speaker repeatedly uses the phrase "open subsets" and says membership in τ is what makes a set open in the example.
In this lesson, "open" is defined relative to the chosen topology τ: a subset of X is open exactly when it is one of the members collected in τ. The video does not appeal to distance or metric notions here; openness is determined by membership in the topology.
A fixed set X and topology τ are chosen.
One is asking whether a subset of X belongs to τ.
The board shows “topological space (X, τ)” and “τ = {all open subsets of X}.”
At 01:01–01:19 the speaker summarizes: “So a topology on X is just a set with elements called open subsets that satisfy those three properties.”
A topological space is written as (X, τ), where X is the underlying set and τ is a collection of subsets of X called open subsets. In this clip, τ is presented as the set of all open subsets of X satisfying three axioms.
τ is a collection of subsets of X.
The members of τ are called open subsets.
τ must satisfy the three topology axioms listed on the board.
Condition (1) on the board reads “∅, X are open ✓.”
At 00:16–00:21 the speaker checks the second example and says it contains the empty set and X, so condition one checks.
For τ to be a topology on X, both the empty set ∅ and the entire underlying set X must belong to τ.
Applies to a proposed collection τ of subsets of X.
Both ∅ and X must be elements of τ.
Condition (2) is marked on the board and checked in examples by union arrows.
At 01:13–01:17 the speaker says, “the union of any number of sets needs to be in tau as well.”
The board does not display the full formal indexed union formula in this clip; the statement is given verbally and illustrated by examples.
A topology must be closed under unions of any number of its open sets. In the examples, unions of listed open sets are checked to see whether the result remains in τ.
Each must be a member of τ.
The index collection may be any number of sets, as stated verbally.
The union must also be a member of τ.
Condition (3) is marked on the board and tested by the intersection example {a} ∩ {a, b} = {a} ∈ τ.
At 01:17–01:19 the speaker says, “any finite intersection also resides in tau.”
The board does not display the full formal indexed finite-intersection formula in this clip; the statement is given verbally and illustrated by examples.
A topology must be closed under finite intersections of its open sets. The clip tests this by intersecting two proposed open sets and checking whether the result is still in τ.
The sets being intersected must be members of τ.
Only finite intersections are required by the axiom as stated in the clip.
The intersection result must also be a member of τ.
The board explicitly lists three numbered requirements: (1) X, ∅ are open; (2) the union of any family of open sets is open; (3) the finite intersection of any collection of open sets is open.
The speaker later restates these conditions while checking the example.
A collection τ of subsets of a nonempty set X is a topology on X if and only if it satisfies: (1) X and ∅ are in τ; (2) the union of any family of sets in τ is in τ; (3) the intersection of any finite collection of sets in τ is in τ.
X is nonempty.
τ is a collection of subsets of X.
Condition (2) quantifies over any family of open sets; condition (3) quantifies over any finite collection of open sets.
The board defines and τ={∅,{a},{a,b},X}, then checks (1), (2), and (3), ending with {a}∩{a,b}={a}∈τ and check marks beside the three conditions.
The speaker asks whether this τ forms a topology on X and concludes that all three conditions are verified via inspection.
For , the collection τ={∅,{a},{a,b},X} satisfies the three topology axioms, so (X,τ) is a topological space.
.
τ={∅,{a},{a,b},X}.
The verification concerns all unions from the displayed family and all finite intersections from the displayed family.
The board writes the example , τ={∅,{a},{a,b},X}, then checks (1) ∅,X are open, marks (2), and writes {a}∩{a,b}={a}∈τ before marking (3).
The speaker walks through the three conditions in order and concludes that all are verified by inspection.
The spoken union check gives representative cases rather than enumerating every possible subfamily explicitly; the board still marks condition (2) as satisfied.
Set up the candidate topology exactly as written on the board.
Given example data from the video.
Check axiom (1): both the empty set and the whole space are present in τ.
Direct inspection of the listed members of τ.
Check axiom (2): unions of chosen open sets do not leave the displayed collection. The speaker illustrates with cases such as taking X union {a}, which gives X, and notes that unions among the listed sets stay inside τ.
Inspection of the finite list of possible unions from the displayed τ.
Check axiom (3) by computing a representative finite intersection and observing that the result is still in τ. The speaker adds that other finite intersections behave similarly.
Direct set-intersection computation plus inspection of τ.
Conclude that the displayed collection is indeed a topology on X, so (X,τ) is a topological space.
By the definition of topology stated earlier in the clip.
The example collection τ={∅,{a},{a,b},X} is a valid topology on .
The board shows , τ = {∅, {a}, {a, b}, X}, condition checks (1), (2), (3), and the intersection calculation {a} ∩ {a, b} = {a} ∈ τ.
Checkmarks and arrows visually indicate that the listed conditions are satisfied.
The full arbitrary-union verification is represented by arrows and checkmarks rather than a complete written list of all possible unions.
The example proposes a topology τ on the finite set X.
Written directly on the board.
The first axiom is checked because both the empty set and the whole set appear in τ.
Board condition (1) and visible membership in the listed τ.
A finite intersection of two members of τ is computed and remains in τ.
Written explicitly beneath the first example.
The example is treated as a valid topology because the three topology conditions are checked on the board.
Checkmarks beside conditions (1), (2), and (3).
The collection τ = {∅, {a}, {a, b}, X} is presented as a topology on .
At 00:11–00:15 the speaker asks whether τ is a topology on X for the second example.
At 00:16–00:21 the speaker says the collection contains the empty set and X, so condition one checks.
At 00:22–00:29 the speaker says the union seems to work because {a, b} union {a, c} gives X.
At 00:29–00:34 the speaker says condition two also seems to work but condition three might fail.
The board writes {a, b} ∩ {a, c} = {a} ∉ τ.
At 00:41–00:47 the speaker explains that the intersection is what the sets have in common, namely a, and that this is not in τ.
The conclusion “τ is not a topology on X” is written on the board.
The union check is verbally asserted for the displayed example rather than fully enumerated on screen.
The second example proposes a different collection τ of subsets of the same underlying set X.
Written on the board while the speaker reads out the sets.
The first topology axiom is satisfied because both ∅ and X are listed in τ.
Speaker at 00:16–00:21 says condition one checks; board also shows these members.
The union of the two nontrivial listed open sets gives the whole set X, which is in τ.
Speaker at 00:22–00:29 states that taking {a, b} and unioning it with {a, c} gives X.
The intersection of the two proposed open sets is computed as their common element a.
Speaker at 00:34–00:45 describes intersecting {a, b} with {a, c}; the equation is written on the board.
The resulting singleton {a} is not one of the listed members of τ.
Written directly as {a} ∉ τ on the board.
Because a finite intersection of members of τ fails to remain in τ, the third topology axiom is violated.
Speaker at 00:48–01:01 concludes that this is a finite intersection not in τ, so τ is not a topology on X.
The proposed collection τ = {∅, {a, b}, {a, c}, X} fails the finite-intersection axiom and therefore is not a topology on .
The board writes "Ex " and "τ = {∅, {a}, {a,b}, X}", then verifies the three topology conditions with check marks and the intersection calculation {a}∩{a,b}={a}∈τ.
The speaker introduces this as a quick simple example and asks whether the displayed τ forms a topology on X.
Given and τ={∅,{a},{a,b},X}, determine whether τ is a topology on X.
.
τ={∅,{a},{a,b},X}.
The definition of topology requires checking the three axioms shown earlier.
Verify all three topology axioms for the displayed τ and conclude whether (X,τ) is a topological space.
Axiom (1) holds because both ∅ and X are explicitly listed as members of τ.
Direct membership check in the displayed collection.
Axiom (2) holds because unions of members of this particular τ remain among the listed sets; the speaker illustrates with examples such as X∪{a}=X.
Finite inspection of the possible unions from the displayed τ.
Axiom (3) is checked by computing a representative finite intersection and seeing that the result is still in τ; the speaker notes the other finite intersections also stay in τ.
Set intersection computation plus membership check in τ.
Since all three axioms are satisfied, the candidate collection qualifies as a topology, so the pair (X,τ) is a topological space.
Application of the definition of topology given at the start of the clip.
Yes. τ={∅,{a},{a,b},X} is a topology on .
The verification is the explicit check of the three axioms written on the board: membership of ∅ and X, closure under unions, and closure under finite intersections, including {a}∩{a,b}={a}∈τ.
Near the end the board begins another example with "Ex " and the speaker starts to define a new τ.
The speaker says "One more example. We have capital X ... let's use the same set ... And this time we'll let tau" before the clip ends.
The second topology τ is not fully written or analyzed within the provided 150-second clip.
No conclusion about the second example is available from this fragment alone.
A second example on the same set is started, apparently to define another τ.
.
The speaker announces another example using the same set.
Define a new τ and test whether it is a topology.
Reuse the same underlying set as in the first example.
Stated by the speaker and written on the board.
The clip stops before the new collection is fully displayed or checked.
Incomplete evidence within the provided interval.
Not completed within this clip.
No verification is possible from the supplied fragment because the second τ and its axiom checks are absent.
The board displays Ex , τ = {∅, {a}, {a, b}, X}, with condition checks and {a} ∩ {a, b} = {a} ∈ τ.
The complete union verification is visual and abbreviated rather than fully expanded on screen.
Determine whether τ = {∅, {a}, {a, b}, X} is a topology on .
.
τ = {∅, {a}, {a, b}, X}.
The three topology conditions are listed on the board.
Check whether τ satisfies the topology axioms.
The empty set and the whole set are both present in τ.
Direct inspection of the listed collection.
A displayed finite intersection remains inside τ.
Written explicitly on the board.
The example is treated as satisfying the three topology conditions.
Checkmarks beside conditions (1), (2), and (3).
Yes, τ = {∅, {a}, {a, b}, X} is presented as a topology on .
The board verifies the required memberships and shows the intersection {a} ∩ {a, b} = {a} remaining in τ.
The board displays Ex , τ = {∅, {a, b}, {a, c}, X}, then {a, b} ∩ {a, c} = {a} ∉ τ and “τ is not a topology on X.”
The speaker reads the collection, checks conditions (1) and (2), predicts condition (3) may fail, and concludes τ is not a topology.
The union condition is checked verbally for the relevant displayed union rather than exhaustively enumerated.
Determine whether τ = {∅, {a, b}, {a, c}, X} is a topology on .
.
τ = {∅, {a, b}, {a, c}, X}.
A topology must contain ∅ and X, be closed under arbitrary unions, and be closed under finite intersections.
Test the three topology axioms and decide whether τ is a topology.
The first axiom holds because both required sets are listed.
Speaker at 00:16–00:21 says condition one checks.
The union of the two nontrivial listed sets gives X, which is in τ.
Speaker at 00:22–00:29 states this union gives X.
The intersection of the two listed open sets is the singleton containing their common element a.
Computed on the board and explained verbally.
The singleton {a} is absent from the proposed collection.
Written directly on the board.
Failure of closure under finite intersections disproves the topology claim.
Speaker at 00:48–01:01 concludes τ is not a topology on X.
No, τ = {∅, {a, b}, {a, c}, X} is not a topology on .
The counterexample {a, b} ∩ {a, c} = {a} ∉ τ violates the finite-intersection axiom.
Text is progressively written on a black background in cyan ink.
Blackboard
Cyan text
The definition and its three properties are written step-by-step.
The background remains black.
The ink color remains cyan.
The visual process mirrors the spoken explanation, building the formal definition of a topological space from its constituent parts.
The screen shows a black digital whiteboard with cyan handwritten text defining topology and the three axioms, then adds τ={all open subsets of X} and topological space (X, τ).
Text appears progressively as the lecturer writes each line.
Cyan handwritten definition of topology.
Numbered axioms (1)-(3).
The line τ={all open subsets of X}.
The phrase topological space (X, τ).
The board first shows the general definition and axioms.
Then τ is identified as the collection of open subsets of X.
Finally the pair notation (X, τ) is added to name a topological space.
The background remains a black digital board.
The mathematical content is presented as handwritten text rather than diagrams or graphs.
The visual sequence moves from abstract definition to concrete notation, establishing that a topology is a selected collection of subsets and that a topological space is the pair made from the set and that collection.
The lower part of the board shows Ex and τ={∅,{a},{a,b},X}, with check marks next to the three conditions and the intersection calculation {a}∩{a,b}={a}∈τ.
The example is written step by step and then annotated with checks as each axiom is verified.
Example set .
Candidate topology τ={∅,{a},{a,b},X}.
Check marks for conditions (1), (2), and (3).
Intersection computation {a}∩{a,b}={a}∈τ.
The example data are introduced first.
Condition (1) is marked satisfied.
Condition (2) is discussed and marked satisfied.
Condition (3) is illustrated by an explicit intersection and then marked satisfied.
The same finite collection τ is being tested throughout this segment.
The underlying set X remains {a,b,c} during the first worked example.
The board turns the abstract axioms into a concrete checklist, showing how membership in τ determines openness and how closure under unions and finite intersections is verified in a small finite example.
A black digital whiteboard shows cyan headings at the top, yellow example text below, and handwritten checkmarks, arrows, and equations added over time.
Cyan heading “topological space (X, τ)”
Cyan line “τ = {all open subsets of X}”
Yellow numbered conditions (1), (2), (3)
Yellow first example with and τ = {∅, {a}, {a, b}, X}
Yellow second example with and τ = {∅, {a, b}, {a, c}, X}
Handwritten intersection calculations and conclusion text
The first example is already accompanied by checkmarks and an intersection verification.
The second example’s τ is completed by adding ∅, {a, b}, {a, c}, and X.
Checkmarks are added for condition (1) and condition (2) in the second example.
The failed intersection {a, b} ∩ {a, c} = {a} ∉ τ is written.
The conclusion “τ is not a topology on X” is written.
The underlying set remains in both examples.
The three topology conditions remain listed at the upper right throughout.
The color coding keeps definitions and space notation in cyan and examples/calculations in yellow.
The visual arrangement separates the general definition of a topological space from two concrete finite-set tests, with the second test ending in a written counterexample.
Arrows in the first example point from listed subsets toward X, visually indicating union checks.
The exact full list of all unions is not written out; the arrows serve as an abbreviated visual check.
Subsets {a}, {a, b}, and X in τ
Arrows pointing toward X
Condition (2) checkmark
The arrows emphasize that unions of the displayed open sets land back inside the collection, especially at X.
The collection τ remains {∅, {a}, {a, b}, X}.
The underlying set remains .
The arrows illustrate closure under unions for the first example without enumerating every possible union separately.
The board writes {a, b} ∩ {a, c} = {a} ∉ τ.
The speaker says the intersection is what they have in common, namely a, and that this is not in τ.
The conclusion “τ is not a topology on X” is written afterward.
Sets {a, b} and {a, c}
Intersection symbol ∩
Resulting set {a}
Non-membership symbol ∉
Conclusion sentence
The intersection expression is built step by step until {a} ∉ τ is complete.
The final conclusion is added to the right of the failed calculation.
The two intersected sets remain members of the proposed τ.
The underlying set X remains {a, b, c}.
The written calculation converts the verbal suspicion that condition three might fail into an explicit counterexample.
The word 'open' is written in quotation marks: '"open" subsets'.
Speaker says 'I'll put this in quotes, open subsets'.
Assuming 'open' has its standard metric space meaning (e.g., involving distances).
In general topology, 'open' is a defined term within the context of the specific topology being constructed; it does not inherently rely on a distance function.
The definition says a topology is a collection of "open subsets" without mentioning distance, intervals, or neighborhoods in a metric sense.
The speaker treats openness as membership in τ, especially when saying the sets are open because they are in τ.
Learners may assume "open set" always means an interval-like or distance-based open set from calculus or analysis.
In this video, openness is defined axiomatically: a subset of X is open precisely when it belongs to the chosen topology τ. The finite example shows sets like {a} can be open even though no metric is mentioned.
The axioms distinguish (2) "the union of any family of open sets is open" from (3) "the finite intersection of any collection of open sets is open".
The speaker explicitly contrasts arbitrary unions with finite intersections while checking the example.
One might think the axioms require both arbitrary unions and arbitrary intersections to stay open.
The video states closure under arbitrary unions but only under finite intersections. In the example, the checked intersection is a finite one: {a}∩{a,b}={a}.
The speaker says the union seems to work and condition two also seems to work, but condition three might fail.
The later written counterexample {a, b} ∩ {a, c} = {a} ∉ τ shows failure despite earlier checks.
One might think that because ∅ and X are present and a displayed union works, the collection is automatically a topology.
All three axioms must hold. In the second example, the union check succeeds, but the finite intersection {a, b} ∩ {a, c} = {a} is not in τ, so τ is not a topology.
At 01:17–01:19 the speaker states that any finite intersection also resides in τ.
The tested expression uses the intersection of two sets, {a, b} ∩ {a, c}.
The clip does not discuss infinite intersections, so the boundary of the axiom is only stated as finite.
One might conflate the union axiom with the intersection axiom and expect closure under arbitrary intersections.
The clip states closure under arbitrary unions but only closure under finite intersections. The counterexample uses a finite intersection of two sets.
Properties (1), (2), and (3) are listed under the main definition.
The first property is a necessary condition contained within the definition of a topology.
Property (2) is listed under the main definition.
The second property (arbitrary unions) is a necessary condition contained within the definition of a topology.
Property (3) is listed under the main definition.
The third property (finite intersections) is a necessary condition contained within the definition of a topology.
After defining τ as a topology, the board writes "topological space (X, τ)".
The speaker says a topological space is the pair consisting of the set and τ.
The notion of a topological space is built directly from the prior definition of a topology on X.
The example checks the same three numbered conditions listed in the definition.
The speaker asks whether the displayed τ forms a topology and then verifies the axioms one by one.
The worked example applies the abstract definition of topology to a concrete finite set and collection.
The definition calls τ a collection of open subsets, and the example labels members of τ as open.
The speaker repeatedly equates being open with belonging to τ.
The terminology of open subsets is contained inside the axiomatic definition: the axioms specify which collections of subsets may serve as the open sets.
The proof on the board checks exactly the three listed axioms for the example τ.
The speaker says all three conditions are verified via inspection.
The derivation depends on the axiom statement: the example is accepted as a topology only because it satisfies those three conditions.
The summary at 01:13–01:17 names the union condition as part of the definition of a topology.
Union checks are illustrated in both examples.
The second topology axiom applies the set-union operation to arbitrary collections of open sets and requires the result to remain in τ.
The summary at 01:17–01:19 names finite intersections as part of the definition of a topology.
The failed check uses {a, b} ∩ {a, c} = {a} ∉ τ.
The third topology axiom applies the set-intersection operation to finitely many open sets and requires the result to remain in τ.
The board writes {a, b} ∩ {a, c} = {a} ∉ τ and then “τ is not a topology on X.”
The speaker concludes that because this finite intersection is not in τ, τ is not a topology on X.
The invalid example depends on the finite-intersection axiom: violating that single axiom is enough to prove the collection is not a topology.
The first example displays checks for the three conditions and the intersection {a} ∩ {a, b} = {a} ∈ τ.
The union verification is abbreviated visually rather than fully enumerated.
The first example applies the definition of a topological space by testing a concrete collection against the three required properties.
The entire segment is dedicated to defining a topology.
Quotation marks around 'open'.
The opening board text defines topology by three axioms.
The speaker introduces the definition before examples.
The board writes "topological space (X, τ)".
The speaker says a topological space is a pair.
The definition calls τ a collection of open subsets of X.
The speaker says sets are open because they are in τ.
The board verifies τ={∅,{a},{a,b},X} against the three axioms.
The speaker asks whether this τ forms a topology and checks each condition.
Axiom (2) mentions any family union and axiom (3) mentions finite intersection.
The speaker contrasts unions of any number of open sets with finite intersections.
A second example begins with but the new τ is not completed onscreen in this clip.
The speaker says "One more example" and starts to define τ before the clip ends.
The full second example lies outside the provided interval.
The board writes “topological space (X, τ)” and “τ = {all open subsets of X}.”
The closing summary defines a topology on X as a set of open subsets satisfying three properties.
The board lists conditions (1), (2), and (3).
At 01:09–01:19 the speaker recites the three properties.
The board writes {a, b} ∩ {a, c} = {a} ∉ τ.
The speaker says this finite intersection is not in τ, so τ is not a topology on X.
The speaker distinguishes the union condition from the finite-intersection condition while checking the second example.
The union {a, b} ∪ {a, c} = X is discussed, while the intersection {a, b} ∩ {a, c} = {a} fails membership.
Covered · The entire clip covers the definition of a topological space and its three axioms.
Covered · General definition of topology, the three axioms, and the pair notation for a topological space.
Covered · Worked finite example , τ={∅,{a},{a,b},X}, with verification of all three topology axioms.
Covered · A second example on the same set is announced and begun, but the new τ is not completed within this clip.
Covered · Opening board state shows the definition of a topological space, the three conditions, and the first valid finite example with its checks.
Covered · The second example is read out and the speaker checks that ∅ and X are present and that the displayed union gives X.
Covered · The speaker tests finite intersections, writes {a, b} ∩ {a, c} = {a} ∉ τ, and concludes τ is not a topology on X.
Covered · The speaker summarizes the definition of a topology on X as a collection of open subsets satisfying the three listed properties.
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