Lecture 15.9 - Path-Connectedness

1. Formula
   Observation

   The handwritten text defines a metric space as S.

2. Audio
   Observation

   The speaker says, 'given a metric space S'.

S

An arbitrary metric space.

Metric space

1. Formula
   Observation

   The handwritten text uses the Greek letter gamma to denote a path function.

2. Audio
   Observation

   The speaker says, 'a continuous function gamma'.

$\gamma$

A continuous function representing a path.

Function from [a,b] to S or E

1. Formula
   Observation

   The handwritten text shows the domain of the path function as [a,b].

2. Audio
   Observation

   The speaker says, 'from a closed interval into the space S'.

[a,b]

A closed interval serving as the domain for a path function.

Subset of real numbers

1. Formula
   Observation

   The handwritten text introduces a subset E of S.

2. Audio
   Observation

   The speaker says, 'we say E subset S is path connected'.

E

A subset of the metric space S.

Subset of S

1. Formula
   Observation

   The handwritten text uses s and t as points in E.

2. Audio
   Observation

   The speaker says, 'for all pairs of points in E'.

s, t

Arbitrary points within the subset E.

Elements of E

1. Formula
   Observation

   The symbol S is written on the whiteboard as part of the phrase 'Given a metric space S' and in the function definition gamma: [a,b] -> S.

S

A metric space

Set of points with a defined distance function

1. Formula
   Observation

   The symbol gamma is written on the whiteboard as part of the function definition gamma: [a,b] -> S and in the conditions gamma(a) = s, gamma(b) = t.

$\gamma$

A continuous function representing a path

[a, b]

1. Formula
   Observation

   The symbols a and b are written on the whiteboard as the domain of the function gamma: [a,b] -> S.

[a, b]

A closed interval of real numbers

$\mathbb{R}$

1. Formula
   Observation

   The symbol E is written on the whiteboard as part of the phrase 'We say $E \subseteq S$ is path-connected' and in the theorem statement.

E

A subset of the metric space S

Subset of S

1. Formula
   Observation

   The symbols s and t are written on the whiteboard as elements of E and as the endpoints of the path gamma.

s, t

Points in the subset E

Elements of E

1. Formula
   Observation

   The symbols $U_1$ and $U_2$ are written on the whiteboard as part of the proof setup: 'Suppose E disconnected by $U_1$, $U_2$'.

$U_1$, $U_2$

Open sets that disconnect the set E

Subsets of the metric space

1. Formula
   Observation

   E is written in the theorem statement and proof.

E

A subset of a topological space S.

Subset of S

1. Formula
   Observation

   Given a metric space S, a path in S is a continuous function $\gamma$: [a,b] $\rightarrow S$.

2. Audio
   Observation

   The speaker states the definition of a path in a metric space.

In a metric space S, a path is defined as any continuous function mapping a closed interval [a,b] into the space S.

1. S is a metric space

2. [a,b] is a closed interval

3. $\gamma$ is a continuous function

1. Formula
   Observation

   We say $E \subseteq S$ is path-connected if $\forall s$, $t \in E$, $\exists \gamma$: [a,b] $\rightarrow E$.

2. Audio
   Observation

   The speaker states the definition of a path-connected set.

A subset E of a metric space S is path-connected if for every pair of points s and t in E, there exists a path entirely contained within E that connects them.

1. E is a subset of a metric space S

2. For all pairs of points s, t in E

3. There exists a continuous function $\gamma$ mapping [a,b] to E

1. Definition of a Path

1. Formula
   Observation

   The whiteboard shows the definition: 'Given a metric space S, a path in S is a ct's function $\gamma$: [a,b] $\to S$.'

2. Audio
   Observation

   The speaker says, 'Given a metric space S, a path in S is a continuous function gamma from a, b to S.'

In a metric space $S$, a path is defined as a continuous function $\gamma$ mapping a closed interval $[a, b]$ of real numbers into $S$. The speaker emphasizes that the function must be continuous.

1. S is a metric space

2. $\gamma$ is a continuous function

3. [a, b] is a closed interval in $\mathbb{R}$

1. Formula
   Observation

   The whiteboard shows: 'We say $E \subseteq S$ is path-connected if $\forall s$, $t \in E \exists \gamma$: [a,b] $\to E s$.t. $\gamma(a)=s$, $\gamma(b)=t$.'

2. Audio
   Observation

   The speaker explains, 'We say E is a subset of S is path connected if for all s, t in E, there exists a gamma from a, b to E such that gamma of a equals s and gamma of b equals t.'

A subset $E$ of a metric space $S$ is path-connected if for any two points $s$ and $t$ in $E$, there exists a path $\gamma$ entirely within $E$ that starts at $s$ and ends at $t$. Specifically, $\gamma(a) = s$ and $\gamma(b) = t$.

1. $E \subseteq S$

2. For every pair of points s, t in E

3. The path gamma maps into E

1. Definition of a Path

1. Formula
   Observation

   The whiteboard shows: 'Thm If $E \subseteq S$ is path-connected, then E is connected.'

2. Audio
   Observation

   The speaker states, 'If E is path connected, then E is connected.'

The theorem states that if a subset $E$ of a metric space $S$ is path-connected, then it is necessarily connected. The speaker initially writes the converse but corrects himself.

1. $E \subseteq S$

2. E is path-connected

1. Definition of Path-Connected Set

1. Formula
   Observation

   The theorem states 'If $E \subseteq S$ is path-connected'.

A set E is path-connected if for any two points in E, there exists a continuous path entirely within E connecting them.

1. E must be a subset of a topological space S.

1. Formula
   Observation

   The theorem concludes 'then E is connected'.

A set E is connected if it cannot be represented as the union of two disjoint, non-empty open subsets relative to E.

1. E must be a subset of a topological space S.

1. Formula
   Observation

   The proof starts with 'Suppose E disconnected by $U_1$, $U_2$'.

A set E is disconnected if there exist two open sets $U_1$ and $U_2$ such that E is contained in their union, E intersects both, and E does not intersect their intersection.

1. $U_1$ and $U_2$ are open sets in the ambient space S.

1. Formula
   Observation

   The text 'the closed topologist's sine curve in $\mathbb{R}^2$' is written.

A specific subset of the two-dimensional Euclidean space used as an example of a connected but not path-connected set.

1. Defined in $\mathbb{R}^2$.

1. Formula
   Observation

   The title on the whiteboard reads 'Example of a connected but not path-connected set: the closed topologist's sine curve in $\mathbb{R}^2$'.

2. Audio
   Observation

   The instructor explains that the set consists of the graph of $f(x) = \sin(1/x)$ for $x > 0$ plus the interval $[-1, 1]$ on the y-axis.

The closed topologist's sine curve is a specific subset of $\mathbb{R}^2$ constructed by taking the graph of the function $f(x) = \sin(1/x)$ for positive $x$, and adding the vertical line segment from $y=-1$ to $y=1$ on the y-axis (where $x=0$).

1. Considered as a subset of $\mathbb{R}^2$ with the standard topology.

1. Audio
   Observation

   The instructor states, 'as a subset of $\mathbb{R}^2$, we're using the graph... It's not just the function... The actual set we're talking about is the graph of the function in $\mathbb{R}^2$.'

In the context of topological properties like connectedness, the object of study is the graph of the function (a set of points $(x, y)$ in $\mathbb{R}^2$), not the function itself (a mapping from a domain to a codomain).

1. Applies when discussing subsets of $\mathbb{R}^2$ defined by functions.

1. Audio
   Observation

   The speaker says, 'And perhaps surprisingly, it doesn't go the other way. So there are actually connected spaces which are not path connected.'

The converse of the theorem is false; there exist connected sets that are not path-connected.

1. E is connected

Existential (there are)

1. Formula
   Observation

   Theorem 22.5 is written on the board: 'If $E \subseteq S$ is path-connected, then E is connected.'

If a subset E of a topological space S is path-connected, then E is connected.

1. $E \subseteq S$

2. E is path-connected

For all subsets E of a topological space S.

1. Audio
   Observation

   The speaker mentions 'our theorem about images of connected sets under continuous functions'.

1. The exact statement of this previous theorem is not written on the board in this clip.

The image of a connected set under a continuous function is connected.

1. The domain set is connected.

2. The function is continuous.

For all connected sets and continuous functions.

1. Formula
   Observation

   The title explicitly states 'Example of a connected but not path-connected set'.

2. Audio
   Observation

   The instructor concludes, 'it turns out that it's connected, but it's not path-connected'.

The closed topologist's sine curve is a connected set, but it is not a path-connected set.

1. The set is defined as the union of the graph of $f(x) = \sin(1/x)$ for $x > 0$ and the interval $[-1, 1]$ on the y-axis.

Universal for the defined set.

1. Formula
   Observation

   The whiteboard shows the start of a proof by contradiction: 'PF Suppose E disconnected by $U_1$, $U_2$. Then $\exists s \in U_1$, $t \in U_2$'.

2. Audio
   Observation

   The speaker outlines the proof strategy: 'Suppose E is disconnected by U1 and U2... we'll just show that that's impossible because if you do that... you can take a point in one piece...'

1. The proof is cut off before completion.

1. Expression

   $$\text{Suppose } E \text{ is disconnected}$$
   Explanation

   Assume for the sake of contradiction that $E$ is disconnected.

   Justification

   Proof by contradiction setup.

   Shown in the video
2. Expression

   $$\text{by } U_1, U_2$$
   Explanation

   By definition of disconnectedness, there exist non-empty, disjoint open sets $U_1$ and $U_2$ (relative to $S$) such that $E \subseteq U_1 \cup U_2$ and $E \cap U_1 \neq \emptyset$, $E \cap U_2 \neq \emptyset$.

   Justification

   Definition of a disconnected set.

   Shown in the video
3. Expression

   $$\exists s \in U_1, t \in U_2$$
   Explanation

   Choose a point $s$ in $E \cap U_1$ and a point $t$ in $E \cap U_2$.

   Justification

   Existence of points in the non-empty intersections.

   Shown in the video

The proof is incomplete in the video, but the setup aims to show that a path between $s$ and $t$ would have to cross the gap between $U_1$ and $U_2$, contradicting continuity or the definition of the separation.

1. Formula
   Observation

   The proof steps are written sequentially on the whiteboard.

1. Expression

   $$\text{Suppose } E \text{ disconnected by } U_1, U_2.$$
   Explanation

   Assume for contradiction that E is not connected.

   Justification

   Start of proof by contradiction.

   Shown in the video
2. Expression

   $$\text{Then } \exists s \in U_1, t \in U_2.$$
   Explanation

   By definition of disconnection, there must be points of E in both $U_1$ and $U_2$.

   Justification

   Definition of disconnected set.

   Shown in the video
3. Expression

   $$\text{so } \exists \gamma \text{ connecting } s, t.$$
   Explanation

   Since E is path-connected, there is a path between any two points in E.

   Justification

   Definition of path-connected set.

   Shown in the video
4. Expression

   $$\text{Then } \gamma([a,b]) \subseteq E \text{ is connected by prev. thm}.$$
   Explanation

   The image of the interval [a,b] under the continuous path $\gamma$ is a subset of E and is connected.

   Justification

   Theorem on images of connected sets under continuous functions.

   Shown in the video
5. Expression

   $$\text{but } U_1, U_2 \text{ disconnect } \gamma.$$
   Explanation

   The sets $U_1$ and $U_2$ also disconnect the path $\gamma$ because it contains points in both.

   Justification

   Properties of disconnection and subset relations.

   Shown in the video
6. Expression

   $$\text{Contradiction.}$$
   Explanation

   We have a set $\gamma([a,b])$ that is both connected and disconnected.

   Justification

   Logical contradiction.

   Shown in the video
7. Expression

   $$\text{E is connected.}$$
   Explanation

   Therefore, the initial assumption that E is disconnected must be false.

   Justification

   Conclusion of proof by contradiction.

   Shown in the video

If E is path-connected, then E is connected.

1. Audio
   Observation

   The instructor explains that a path from a point on the oscillating curve to a point on the y-axis interval would require passing through an infinite number of oscillations, which is impossible for a continuous function on a closed interval.

1. Explanation

   Consider a point on the oscillating part of the curve (where $x > 0$) and a point on the vertical segment on the y-axis (where $x = 0$).

   Justification

   Setup for attempting to construct a path.

   Shown in the video
2. Explanation

   Any continuous path connecting these two points must traverse the region where the function $\sin(1/x)$ oscillates infinitely often as $x$ approaches 0.

   Justification

   Visual observation of the graph's behavior near the y-axis.

   Shown in the video
3. Explanation

   Passing through an infinite number of oscillations in finite time (or over a closed interval domain for the path parameter) violates the continuity requirement for paths.

   Justification

   Standard topological argument regarding uniform continuity or bounded variation on compact intervals (implied by the instructor's 'not possible if it's a continuous function on a closed interval').

   Derived from the video

Therefore, no such continuous path exists, meaning the set is not path-connected.

1. Audio
   Observation

   The speaker mentions parametric curves in higher dimensional spaces as examples of paths.

Identify an example of a path in a specific metric space.

1. The metric space is $\mathbb{R}^k$.

Provide an example of a path in $\mathbb{R}^k$.

1. Explanation

   Consider parametric curves in higher-dimensional spaces.

   Justification

   These curves are continuous functions from a closed interval into $\mathbb{R}^k$.

   Shown in the video

Parametric curves in $\mathbb{R}^k$ are examples of paths.

They fit the definition of a continuous function from [a,b] to S where $S = \mathbb{R}^k$.

1. Formula
   Observation

   Text: 'Example of a connected but not path-connected set: the closed topologist's sine curve in $\mathbb{R}^2$'.

2. Diagram
   Observation

   A sketch of the topologist's sine curve is drawn.

Provide an example of a set that is connected but not path-connected.

1. The set is the closed topologist's sine curve in $\mathbb{R}^2$.

Illustrate the distinction between connectedness and path-connectedness.

1. Expression

   $$\text{Sketch the graph of } y = \sin(1/x) \text{ for } x > 0 \text{ and the segment } x=0, -1 \le y \le 1.$$
   Explanation

   Draw the oscillating curve approaching the y-axis and include the y-axis segment to make it closed.

   Justification

   Definition of the closed topologist's sine curve.

   Shown in the video

The closed topologist's sine curve is connected but not path-connected.

The visual sketch shows the set's structure, which intuitively prevents a continuous path from the origin to the oscillating part while maintaining overall connectedness.

1. Animation
   Observation

   The speaker draws a circle around the letter E in the formula $\gamma$: [a,b] $\rightarrow E$.

2. Audio
   Observation

   The speaker says, 'So I'm going to circle that because that's important.'

1. The letter E in the formula

1. A circle is drawn around the letter E.

1. The rest of the formula remains unchanged.

Emphasizes that the path must map into the subset E itself, not just the larger metric space S.

1. Animation
   Observation

   The speaker erases the words 'connected' and 'path-connected' in the theorem statement and rewrites them in the reverse order.

2. Audio
   Observation

   The speaker says, 'Or sorry, sorry. I got this flipped. Ay ay ay.'

1. Text on whiteboard

1. 'connected' and 'path-connected' swap positions in the implication.

1. The structure 'If $E \subseteq S$ is ..., then E is ...' remains.

The speaker realized he wrote the converse of the intended theorem and corrected it.

1. Animation
   Observation

   Handwritten text appears sequentially on the whiteboard, forming the proof.

1. Whiteboard

2. Handwritten text

1. Text is added line by line to construct the proof.

1. The camera angle remains fixed on the whiteboard and the speaker.

The sequential writing guides the viewer through the logical steps of the proof by contradiction.

1. Animation
   Observation

   A hand draws a coordinate system and then sketches the topologist's sine curve.

1. Coordinate axes

2. Oscillating curve

3. Vertical line segment

1. Axes are drawn first, followed by the curve and the vertical segment.

1. The drawing is a freehand sketch without precise mathematical plotting.

The sketch visually represents the set described in the example, showing its oscillatory behavior near the y-axis.

1. Animation
   Observation

   The instructor draws the axes and then sketches the oscillating curve $y = \sin(1/x)$ for $x > 0$, showing increasing frequency as $x$ approaches 0.

1. x-axis

2. y-axis

3. Oscillating curve

1. Curve is drawn from right to left, getting denser near the y-axis.

1. Amplitude remains bounded between -1 and 1.

Visual representation of the function $f(x) = \sin(1/x)$ for $x > 0$.

1. Animation
   Observation

   The instructor shades/draws a thick vertical line segment on the y-axis between -1 and 1 and labels it 'add interval [-1, 1] on y-axis'.

1. y-axis segment from -1 to 1

1. Vertical segment is emphasized/added to the drawing.

1. Position on the y-axis ($x=0$).

Completing the definition of the closed topologist's sine curve by including the limit points on the y-axis.

1. Audio
   Observation

   The speaker emphasizes circling the E because it's important that the path is in E.

Believing that a path connecting two points in a subset E only needs to map into the larger metric space S.

For E to be path-connected, the path must map entirely into the subset E, not just the ambient space S.

1. Audio
   Observation

   The speaker initially writes 'If E is connected, then E is path-connected' before correcting it.

One might assume that connectedness implies path-connectedness.

The speaker corrects this, stating that path-connectedness implies connectedness, but not vice versa.

1. Audio
   Observation

   The speaker introduces an example of a connected but not path-connected set immediately after proving the forward implication.

Assuming that if a set is connected, it must also be path-connected.

The closed topologist's sine curve is a standard counterexample showing that connectedness does not imply path-connectedness.

1. Audio
   Observation

   The instructor explicitly warns, 'It's not just the function... The actual set we're talking about is the graph of the function in $\mathbb{R}^2$.'

Thinking that topological properties like connectedness apply directly to the algebraic expression of the function rather than the geometric set of points forming its graph in the plane.

In this context, the object is the subset of $\mathbb{R}^2$ consisting of points $(x, y)$ satisfying the equation, not the mapping $f: (0, \infty) \to [-1, 1]$.

1. Formula
   Observation

   The definition of path-connectedness uses the previously defined concept of a path.

The definition of a path-connected set relies on the prior definition of a path.

1. Formula
   Observation

   The theorem statement 'If E is path-connected, then E is connected' establishes a logical implication.

The definition of path-connectedness is the hypothesis of the theorem.

1. Audio
   Observation

   The speaker explicitly contrasts the two concepts by stating the converse is false.

The theorem shows one direction of implication, while the claim notes the lack of equivalence.

1. Audio
   Observation

   The instructor uses this example to show a set that is connected but fails to be path-connected.

This example demonstrates that the converse of 'path-connected implies connected' is false; a set can be connected without being path-connected.

1. Formula
   Observation

   The definition of a path is written on the board.

1. Definition of a Path

1. Formula
   Observation

   The definition of path-connectedness is written on the board.

1. Definition of Path-Connectedness

1. Animation
   Observation

   The speaker circles the E in the formula.

1. Definition of Path-Connectedness
2. Path Target Space

1. Formula
   Observation

   Definition of a path is written on the board.

1. Definition of a Path

1. Audio
   Observation

   Speaker discusses the theorem and its converse.

1. Theorem: Path-Connected implies Connected
2. Converse is False

1. Formula
   Observation

   The entire proof is written on the board.

1. Path-connected implies connected
2. Proof that path-connected implies connected

1. Formula
   Observation

   The example is written and sketched.

1. Closed topologist's sine curve
2. Closed topologist's sine curve

1. Formula
   Observation

   Title: 'Example of a connected but not path-connected set'.

1. Closed Topologist's Sine Curve
2. Topological Properties of the Closed Topologist's Sine Curve

1. Audio
   Observation

   Explanation involving infinite oscillations preventing a continuous path.

1. Intuitive Derivation of Non-Path-Connectedness