Introducing Homeomorphisms with a meme

Notes

This was a beamer slideshow that [badly] converted into a more text like format. This was also meant for a "maths talk" I did on how to introduce homeomorphisms in a more interesting way.

Introducing Continuity

Continuity: The idea of continuity isn't well defined before analysis, and the definition changes. For example many people are told that a function is continuous if it can be graphed without picking up your pen. Topology offers us a much clearer definition of continuity:

Definition: A function f from a topological space X to Y is continuous if for every $V \subseteq Y$, $f^{-1}(V)$ is open in X

Donut = Cup?

Why Donut = Cup? The infamous meme that claims topologist think that a donut is the same as a mug or cup. Well, the reality is that that is true, except possibly not in the way you would expect. The topological properties of the cup/mug and a torus(donut) are the same, so we introduce a concept called a homeomorphism.

Homeomorphisms: A homeomorphism is very similar to a isomorphism from algebra, but where instead of preserving algebraic properties, the homeomorphism preserves topological properties like genus and compactness.

Definition A homeomorphism is a function $f:X \mapsto Y$, where X and Y are topological spaces, such that:

  1. f is bijective

  2. f is continuous

  3. $f^{-1}$ is continuous

Defining a donut

Introduction: In this next section, I will introduce the quotient topology, so that we can create the donut(torus), and examine some of its topological properties(and maybe even analyze what we can tell about a mug or cup from it). The idea of an equivalence class(and relation) will be an important part in understanding the quotient topology:

Definition: A binary relation, $\sim$, on a set X is said to be an equivalence relation if it satisfies certain conditions(for all $x,y,z \in X$):

  1. $x \sim x$

  2. $x \sim y$ iff $y \sim x$

  3. if $x\sim y$ and $y\sim z$, $x\sim z$

Equivalence Class

Definition: For some $x \in X$, an equivalence class is said to be the subset of X given by := $\{y \in X \mid x \sim y\}$ We can also define the set of all equivalence classes on X as: $(X/ \sim) = \{[x] \mid x\in X\}$

Now, we can define the open set in ($X/\sim$) :

Definition: For some $U \subseteq (X/\sim)$, where X is a topological space and $\sim$ is an equivalence relation on it, U is open iff: $\bigcup_{[x] \subseteq U} [x] \subseteq X$ is open in X

Here, we have created the quotient topology! We can confirm that this is indeed a topology

Creating a Torus: To create a torus, we use the equivalence relation $(x,y)\sim (x+1,y) \sim (x, y+1)$ in the Cartesian plane. This is wonderful, but now we can analyze some of the topological properties of a torus.

Example Property: A torus is topologically compact, and therefore so is a mug. A compact space is one where any covering of the space has a finite subcover.

End Notes

We can use the idea of a quotient map to also build a quotient topology, but for the purpose of creating a torus, I used the method using equivalence classes:

Definition A surjective map $p:X \mapsto Y$, where X and Y are topological spaces, is said to be a quotient map if for $V \subseteq$ Y, V is open in Y iff $f^{-1}(V)$ is open in X

using this idea, we can show that given a set A and a space X, there exists only one topology on A, relative to p that is, and this topology is called the topology induced by p

References

References Topology Second Edition. Munkres, James. Pearson Publishing Quotient Topology Supplement, http://www.math.ucsd.edu/$\sim$bprhoades/190w16/quotient.pdf

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