Topology and Continuous Functions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 2 November 2009

Topology

A topological space is a set $X$ with a specification of the open subsets of $X$ where it is required that

1. $\varnothing$ is open and $X$ is open,
2. Unions of open sets ore open,
3. Finite intersections of open sets are open.

In other words, a topology on $X$ is a set $𝔗$ [ $𝒯$ - 𝒯 IS NOT DISPLAYING CORRECTLY ON MY MACHINE, USING FRACTUR INSTEAD] of subsets of $X$ such that

1. $\varnothing \in 𝔗$ and $X\in 𝔗$,
2. If $U_i\in 𝔗$ , then $\underset{i}{\cup }{U}_i\in 𝔗$,
3. If $U_1,U_2,\dots ,U_n$ is a finite collection of elements of $𝔗$ , then $\underset{i}{\cap }{U}_i\in 𝔗.$

A topological space is a set $X$ with a topology $𝔗$ on $X$.

Let $𝔗$ be a topology on $X$. An open set is a set in $𝔗$.

A closed set is a subset $E$ of $X$ such that the complement $E^c$ of $E$ is open.

Let $X$ be a topological space and let $x\in X$. A neighbourhood of $x$ is an open subset $U$ of $X$ such that $x\in U$.

Let $X$ be a topological space. A subspace of $X$ is a subset $Y$ of $X$ with the topology given by making the open sets be the sets $${\iota }^{-1}\left(V\right)\text{such that}V\text{is an open subset of}X\text{,}$$ where $\iota :Y\to X$ is the inclusion.

Continuous Functions

Continuous functions are for comparing topological spaces.

Let $X$ and $Y$ be topological spaces. A function $f:X\to Y$ is continuous if it satisfies the condition $$\text{if}V\text{is an open subset of}Y\text{then}{f}^{-1}\left(V\right)\text{is an open subset of}X\text{.}$$

Let $X$ and $Y$ be topological spaces. Let $a\in X$. A function $f:X\to Y$ is continuous at $a$ if it satisfies the condition $$\text{if}V\text{is a neighbourhood of}f\left(a\right)\text{in}Y\text{then}{f}^{-1}\left(V\right)\text{is a neighbourhood of}a\text{in}X\text{.}$$

Let $X$ and $Y$ be topological spaces and let $a\in X$. A function $f:X\to Y$ is continuous at $a$ if and only if $\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$.

Let $X$ and $Y$ be topological spaces. An isomorphism or homeomorphism is a continuous function $f:X\to Y$ such that the inverse function $f^{-1}:Y\to X$ exists and is continuous.

Let $f:X\to Y$ be a continuous function. If $X$ is connected then $f\left(X\right)$ is connected.

		Proof.
	Proof by contradiction. Assume $f\left(X\right)$ is not connected. Let $A$ and $B$ be open in $f\left(X\right)$ such that $A\cup B=f\left(X\right)$ and $A\cup B=\varnothing$. Then let $C=f^{-1}\left(A\right)$ and $D=f^{-1}\left(B\right)$. Then $C\cup D=f^{-1}\left(A\right)\cup f^{-1}\left(B\right)=f^{-1}\left(A\cup B\right)=f^{-1}\left(f\left(X\right)\right)=X$, and $C\cap D=f^{-1}\left(A\right)\cap f^{-1}\left(B\right)=f^{-1}\left(A\cap B\right)=f^{-1}\left(\varnothing \right)=\varnothing$. Now $C\ne \varnothing$ since $A\ne \varnothing$ and $A\subseteq f\left(X\right)$, and $D\ne \varnothing$ since $B\ne \varnothing$ and $B\subseteq f\left(X\right)$. So $X$ is not connected. This is a contradiction. So $f\left(X\right)$ is connected.

Examples

Let $X$ be a set. The discrete topology on $X$ is the topology such that every subset of $X$ is open.

A metric space is a set $X$ with a function $d:X\times X\to {ℝ}_{\ge 0}$ such that

1. If $x\in X$ then $d\left(x,x\right)=0$,
2. If $x,y\in X$ and $d\left(x,y\right)=0$, then $x=y$,
3. If $x,y,z\in X$ then $d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right)$.

Let $X$ be a metric space. Let $x\in X$ and let $ϵ\in {ℝ}_{>0}$. The ball of radius $ϵ$ at $x$ is the set $$B_{ϵ}\left(x\right)=\left\{p\in X|d\left(x,y\right)\le ϵ\right\}\text{.}$$

Let $X$ be a metric space. The metric space topology on $X$ is the topology generated by the sets

$B_{ϵ}\left(x\right)\text{,}$ for $x\in X$ and $ϵ\in {ℝ}_{>0}$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)