Classnotes for September 15

From 0506Topology

#	Week of...	Links (edit)
Fall
1	Sep 12	About, Tue, Thu, Std2Disc
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6	Oct 17	Tue, Thu, HW3
7	Oct 24	Mon, Tue, Thu
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20	Feb 27	Tue, Thu, HW9
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Study	Apr 17	Office Hours
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Classnotes for September 15

Table of contents

1 Topology Riddle

2 Basic Properties of Open Sets

3 Topological Spaces

4 Continuity

5 Scanned Notes

5.1 JPG, by Annat Koren
5.2 PDF, by Gabriel Lee

6 One draw-based solution for the Topology Riddle.

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Topology Riddle

Theorem: Every two-dimensional orientable surface (with minor conditions) is toplogically equivalent to a sphere with $g$ tunnels and $n$ punctures for some $g, n$ .

See left for an example ( $g = 4, n = 3$ ).

Riddle:

Oh really? How about the manifold pictured left?

What are the values of $g$ and $n$ , and how can it be deformed into an object of the above format?

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Basic Properties of Open Sets

We begin by reviewing the definitions of an open subset of $\mathbb{R}^n$ and continuous functions from $\mathbb{R}^m$ to $\mathbb{R}^n$ with these hopes of generalizing the notions to more general spaces.

Definition: A set $U \subset \mathbb{R}^n$ is said to be open if $\forall x \in U \ \exists \epsilon > 0$ s.t. $B_\epsilon(x) \subset U$ .

Theorem: A function $f \colon \mathbb{R}^m \to \mathbb{R}^n$ is continuous iff. $\forall$ open $U \subset \mathbb{R}^n \ f^{-1}(U)$ is also open.

Open sets as defined above possess three basic properties, which are established by the following theorem:

Theorem:

$\emptyset$ and $\mathbb{R}^n$ are open.
If $I$ is an index set (of arbitrary cardinality), and if $\forall \alpha \in I \ U_\alpha$ is open, then $\bigcup_{\alpha\in I} U_\alpha$ is open.
If $I$ is a finite index set, and if $\forall i \in I \ U_i$ is open, then $\bigcap_{i\in I} U_i$ is open.

Proof: The first part of the theorem is vacuously true, and the proof of the second is a triviality, so we shall focus on the third part. Let $I$ and the $U i$ be as in the statement of this part of the theorem, and let $x \in U = \bigcap_{i\in I} U_i$ . By definition, then, $\forall i \in I \ x \in U_i$ . By the openness of each $U i$ , $\forall i \in I \ \exists \epsilon_i > 0$ such that $B_{\epsilon_i}(x) \subset U_i$ . Let $ε$ be the minimum of all the $ε i$ , which exists since $I$ is finite, and is thus positive since all the $ε i$ are positive. It follows that $\forall i \in I \ B_\epsilon(x) \subset B_{\epsilon_i}(x) \subset U_i$ . Thus $B_\epsilon(x) \subset \bigcap_{i\in I} U_i$ , and so, $\bigcap_{i\in I} U_i$ is open, q.e.d.

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Topological Spaces

We can now define our more general space: one which has subsets we can rightfully call "open" and which we can therefore use to define continuity in the most general setting possible. Let us remind ourselves that if $X$ is a set, then $\mathcal{P}\left (X \right )$ (alternatively $2 X$ ) denotes the power set of $X$ , the set of all subsets of $X$ .

Definition: A topological space is a pair $(X, \mathcal{T})$ where $X$ is a set and $\mathcal{T} \subset \mathcal{P}(X)$ such that:

$\emptyset , X \in \mathcal{T}$ .
If $I$ is an index set (of arbitrary cardinality), and if $\forall \alpha \in I \ U_\alpha \in \mathcal{T}$ then $\bigcup_{\alpha\in I} U_\alpha \in \mathcal{T}$ .
If $I$ is a finite index set, and if $\forall i \in I \ U_i \in \mathcal{T}$ , then $\bigcap_{i\in I} U_i \in \mathcal{T}$ .

$\mathcal{T}$ is called a topology on $X$ , and the elements of $\mathcal{T}$ are called open sets.

Some examples of topological spaces as defined above are:

The so-called standard topology on $\mathbb{R}^n$ , given by $\mathcal{T}_{std} = \{U \subset \mathbb{R}^n \colon U$ is open according to the old definition from analysis $}$ .
The trivial topology on any set $X$ , given by $\mathcal{T}_{tr} = \{\emptyset, X\}$ .
The discrete topology on any set $X$ , given by $\mathcal{T}_{dsc} = \mathcal{P}(X)$ . By definition, every subset of $X$ is open in this topology.

From now on we follow the general practice (strictly speaking an abuse of notation) of denoting a topological space $(X, \mathcal{T})$ by $X$ , suppressing the topology (cf. the parallel practice of suppressing the multiplication when referring to a group). Should the need arise to refer explicitly to the topology of a topological space denoted by, say, $Y$ , we can denote it by $\mathcal{T}_Y$ .

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Continuity

We now generalize the defintion of continuity to work for topological spaces:

Definition: If $X$ and $Y$ are topological spaces and $f \colon X \to Y$ , we say that $f$ is continuous if $f^{-1}(U) \in \mathcal{T}_X$ whenever $U \in \mathcal{T}_Y$ .

This new definition of continuity still admits many of the same properties that the old definition from analysis, upon which it was modelled, does.

Theorem:

If $X$ is a topological space, then the identity function $id \colon X \to X$ is continuous.
If $X, Y, Z$ are topological spaces and $f \colon X \to Y$ , $g \colon Y \to Z$ are continuous, it follows that $g \circ f \colon X \to Z$ is continuous.

Proof:

Since $i d = i d - 1$ , it follows that $i d - 1 (U) = i d (U) = U$ . Hence, if $U \subset X$ is open, then $I - 1 (U)$ is trivially open too.
It is easy to show that $(g\circ f)^{-1}(U) = f^{-1}(g^{-1}(U))$ . Hence, if $U \subset X$ is open, then $g - 1 (U)$ is open by the continuity of $g$ , and thus $(g\circ f)^{-1}(U) = f^{-1}(g^{-1}(U))$ is open by the continuity of $f$ , q.e.d.

Since every set admits at least two different topologies (i.e. the discrete and the trivial topologies), it is natural to ask, for instance, what happens to the continuity of a function from a set to itself when different topologies are assigned to the domain and codomain.

So, let us consider the case of the real line $\mathbb{R}$ , together with the standard topology $\mathcal{T}_{std}$ , the trivial topology $\mathcal{T}_{triv}$ , and the discrete topology $\mathcal{T}_{dsc}$ , and in particular, examine the continuity of the identity function $id \colon \mathbb{R} \to \mathbb{R}$ when various topologies are assigned to the domain and the codomain. We summarise our results in the following table:

			Domain
		$\left ( \mathbb{R},\mathcal{T}_{triv} \right )$	$\left ( \mathbb{R},\mathcal{T}_{std} \right )$	$\left ( \mathbb{R},\mathcal{T}_{dsc} \right )$
	$\left ( \mathbb{R},\mathcal{T}_{triv} \right )$	continuous;	continuous;	continuous;
Codomain	$\left ( \mathbb{R},\mathcal{T}_{std} \right )$	not continuous;	continuous;	continuous;
	$\left ( \mathbb{R},\mathcal{T}_{dsc} \right )$	not continuous;	not continuous;	continuous;

We can repeat this analysis for functions in general from $\mathbb{R}$ to itself, identifying which functions are continuous in each case:

			Domain
		$\left ( \mathbb{R},\mathcal{T}_{triv} \right )$	$\left ( \mathbb{R},\mathcal{T}_{std} \right )$	$\left ( \mathbb{R},\mathcal{T}_{dsc} \right )$
	$\left ( \mathbb{R},\mathcal{T}_{triv} \right )$	every function;	every function;	every function;
Codomain	$\left ( \mathbb{R},\mathcal{T}_{std} \right )$	constant functions only;	continuous (old defintion) functions;	every function;
	$\left ( \mathbb{R},\mathcal{T}_{dsc} \right )$	constant functions only;	constant functions only;	every function;

These results can be checked with the tools that we already have except for the result that a function $f \colon \left ( \mathbb{R},\mathcal{T}_{std} \right ) \to \left ( \mathbb{R},\mathcal{T}_{dsc} \right )$ is continuous iff. it is a constant function. This particular result will be proved later on.

Some of the results from the above tables can be generalised quite readily:

Proposition: Let $X$ be a set, let $\mathcal{T}_{triv}$ and $\mathcal{T}_{dsc}$ be the trivial and discrete topologies on $X$ , respectively, and let $\mathcal{T}$ be any topology on $X$ . Then:

Every function $f \colon \left ( X, \mathcal{T} \right ) \to \left ( X, \mathcal{T}_{triv} \right )$ is continuous.
Every function $f \colon \left ( X, \mathcal{T}_{dsc} \right ) \to \left ( X, \mathcal{T} \right )$ is continuous.

Proof:

Continuity follows from the fact that $f^{-1} \left ( \empty \right ) = \empty$ and $f^{-1} \left ( X \right ) = X$ .
Continuity follows from the fact that every subset of $X$ is open in the discrete topology, q.e.d.

Thus, for any given set, the discrete topology can be said to be the "strongest" topology, being stronger than all other possible topologies, and the trivial topology can be said to be the "weakest" topology, being weaker than all other possible topologies.

We end this lesson with some terminology:

Definition: Let $X$ be a set, and let $\mathcal{T}_1$ and $\mathcal{T}_2$ be topologies on $X$ such that $\mathcal{T}_2 \subset \mathcal{T}_1$ . Then:

$\mathcal{T}_1$ is said to be stronger, bigger, or finer than $\mathcal{T}_2$ .
$\mathcal{T}_2$ is said to be weaker, smaller, or coarser, respectively, than $\mathcal{T}_1$ .

Using this language we can state a generalisation of the proposition we just proved:

Proposition: Let $X$ be a set, and let $\mathcal{T}_1$ and $\mathcal{T}_2$ be topologies on $X$ , with $\mathcal{T}_1$ stronger than $\mathcal{T}_2$ . Then the identity function $id \colon \left ( X, \mathcal{T}_1 \right ) \to \left ( X, \mathcal{T}_2 \right )$ is continuous.