# Topology/Continuity and Homeomorphisms

 Topology ← Quotient Spaces Continuity and Homeomorphisms Separation Axioms →

## Continuity

Continuity is the central concept of topology. Essentially, topological spaces have the minimum necessary structure to allow a definition of continuity. Continuity in almost any other context can be reduced to this definition by an appropriate choice of topology.

### Definition

Let $X,Y$ be topological spaces.

A function $f:X\to Y$ is continuous at $x\in X$ if and only if for all open neighborhoods $B$ of $f(x)$ , there is a neighborhood $A$ of $x$ such that $A\subseteq f^{-1}(B)$ .
A function $f:X\to Y$ is continuous in a set $S$ if and only if it is continuous at all points in $S$ .

The function $f:X\to Y$ is said to be continuous over $X$ if and only if it is continuous at all points in its domain.

$f:X\to Y$ is continuous if and only if for all open sets $B$ in $Y$ , its inverse $f^{-1}(B)$ is also an open set.
Proof:
($\Rightarrow$ )
The function $f:X\to Y$ is continuous. Let $B$ be an open set in $Y$ . Because it is continuous, for all $x$ in $f^{-1}(B)$ , there is a neighborhood $x\in A\subseteq f^{-1}(B)$ , since B is an open neighborhood of f(x). That implies that $f^{-1}(B)$ is open.
($\Leftarrow$ )
The inverse image of any open set under a function $f$ in $Y$ is also open in $X$ . Let $x$ be any element of $X$ . Then the inverse image of any neighborhood $B$ of $f(x)$ , $f^{-1}(B)$ , would also be open. Thus, there is an open neighborhood $A$ of $x$ contained in $f^{-1}(B)$ . Thus, the function is continuous.

If two functions are continuous, then their composite function is continuous. This is because if $f$ and $g$ have inverses which carry open sets to open sets, then the inverse $g^{-1}(f^{-1}(x))$ would also carry open sets to open sets.

### Examples

• Let $X$ have the discrete topology. Then the map $f:X\rightarrow Y$ is continuous for any topology on $Y$ .
• Let $X$ have the trivial topology. Then a constant map $g:X\rightarrow Y$ is continuous for any topology on $Y$ .

## Homeomorphism

When a homeomorphism exists between two topological spaces, then they are "essentially the same", topologically speaking.

### Definition

Let $X,Y$ be topological spaces
A function $f:X\to Y$ is said to be a homeomorphism if and only if

(i) $f$ is a bijection
(ii) $f$ is continuous over $X$ (iii)$f^{-1}$ is continuous over $Y$ If a homeomorphism exists between two spaces, the spaces are said to be homeomorphic

If a property of a space $X$ applies to all homeomorphic spaces to $X$ , it is called a topological property.