# 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.

## Exercises

1. Prove that the open interval $(a,b)$  is homeomorphic to $\mathbb {R}$ .
2. Establish the fact that a Homeomorphism is an equivalence relation over topological spaces.
3. (i)Construct a bijection $f:[0,1]\to [0,1]^{2}$
(ii)Determine whether this $f$  is a homeomorphism.

 Topology ← Quotient Spaces Continuity and Homeomorphisms Separation Axioms →