# 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 ${\displaystyle X,Y}$  be topological spaces.

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

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

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

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

### Examples

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

## Homeomorphism

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

### Definition

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

(i) ${\displaystyle f}$  is a bijection
(ii) ${\displaystyle f}$  is continuous over ${\displaystyle X}$
(iii)${\displaystyle f^{-1}}$  is continuous over ${\displaystyle Y}$

If a homeomorphism exists between two spaces, the spaces are said to be homeomorphic

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

### Notes

1. A map may be bijective and continuous, but not a homeomorphism. Consider the bijective map ${\displaystyle f:[0,1)\rightarrow S^{1}}$ , where ${\displaystyle f(x)=e^{2\pi ix}}$  mapping the points in the domain onto the unit circle in the plane. This is not a homeomorphism, because there exist open sets in the domain that are not open in ${\displaystyle S^{1}}$ , like the set ${\displaystyle \left[0,{\frac {1}{2}}\right)}$ .
2. Homeomorphism is an equivalence relation

## Exercises

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

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