# Homeomorphisms. Homeomorphic spaces.

### Homeomorphisms and homeomorphic spaces.

Let \( X \) and \( Y \) be topological spaces.

- Homeomorphism
- A
*homeomorphism*from \( X \) to \( Y \) is a continuous bijection whose inverse is also continuous. - Homeomorphic spaces
- The spaces \( X \) and \( Y \) are
*homeomorphic*iff there exists a homeomorphism between them.

\( X \cong Y \) is the notation for \( X \) to be homeomorphic to \( Y \). '*Topologically equivalent*' is an alternative term for 'homeomorphic'.

### When are two spaces homeomorphic?

To show that two spaces are homeomorphic, find a homeomorphism between them.

*Roughly*, two spaces are homeomorphic iff one can be deformed into
the other by bending and reshaping, but without tearing or gluing.

### Equivalence relation

Being homeomorphic is an equivalence relation on the class of all topological spaces.

- Reflexive
- Let \( X \) be a topological space. Then the identity map on \( X \) is a homeomorphism.
- Symmetric
- Let \( f : X \to Y \) be a homeomorphism. Then \( f^{-1} : Y \to X \) is a homeomorphism.
- Transitive
- Let \( f : X \to Y \) and \( g : Y \to Z \) be homeomorphisms. Then \( g \circ f : X \to Z \) is a homeomorphism.

Thus, \( X \cong X \); \( X \cong Y \iff Y \cong X \); \( X \cong Y \land Y \cong Z \implies X \cong Z \).

### Knots homeomorphic to the circle

Some cases of homeomorphisms seem to defy the restriction to bending and reshaping: all knots are homeomorphic to the circle. To see this, take two knots of 1m string; choose a point on each as a starting point; then, trace around each knot at the same rate until you reach the start again. This defines a homeomorphism. While we can't bend a knot into a circle in \( \mathbb{R}^3 \), any knot can be reshaped into a circle if both are first placed in \( \mathbb{R}^4 \).

#### Bijection with non-continuous inverse

The inverse of a continuous bijection *need not be
continuous*.

An easy way of constructing an example is to consider continuous maps to a set with the discrete topology. For example, the identity map \( i : (\mathbb{R}, \text{discrete topology}) \to (\mathbb{R}, \text{standard topology}) \) is continuous, but the inverse, \( \) \( i^{-1} : (\mathbb{R}, \text{standard topology}) \to (\mathbb{R}, \text{discrete topology}) \) is not continuous.