Homeomorphisms

A homeomorphism is an isomorphism in the category of topological spaces. Two spaces that are homeomorphic are considered "the same" from a topological viewpoint -- they share all topological properties. Identifying which spaces are homeomorphic is one of the central questions in topology.

Definition

Definition2.3Homeomorphism

A function $f: X \to Y$ between topological spaces is a homeomorphism if:

$f$ is a bijection,
$f$ is continuous, and
$f^{-1}: Y \to X$ is continuous.

If such a function exists, $X$ and $Y$ are said to be homeomorphic, written $X \cong Y$ .

RemarkContinuous Bijection Need Not Be a Homeomorphism

Condition (3) is not redundant. A continuous bijection need not have a continuous inverse.

Consider $f: [0, 2\pi) \to S^1$ defined by $f(t) = (\cos t, \sin t)$ , where $[0, 2\pi)$ has the subspace topology from $\mathbb{R}$ and $S^1 \subseteq \mathbb{R}^2$ . Then $f$ is a continuous bijection, but $f^{-1}$ is not continuous: the image of $[0, \epsilon)$ under $f$ is not open in $S^1$ for any $\epsilon > 0$ .

Examples

ExampleClassical Homeomorphisms

Open intervals and $\mathbb{R}$ : The map $f: (-1, 1) \to \mathbb{R}$ given by $f(x) = \frac{x}{1 - x^2}$ is a homeomorphism. In fact, any two open intervals $(a, b) \cong (c, d)$ via the affine map $x \mapsto c + \frac{(d-c)(x-a)}{b-a}$ .
$(0, 1) \cong \mathbb{R}$ : Via $f(x) = \tan(\pi x - \pi/2)$ .
$\mathbb{R}^n \setminus \{0\} \cong S^{n-1} \times (0, \infty)$ : Via $(x_1, \ldots, x_n) \mapsto \left(\frac{x}{\|x\|}, \|x\|\right)$ .
Stereographic projection: $S^n \setminus \{N\} \cong \mathbb{R}^n$ where $N = (0, \ldots, 0, 1)$ is the north pole.
Coffee cup and donut: The torus $T^2 = S^1 \times S^1$ is homeomorphic to the surface of a coffee mug with one handle -- the classic illustration that topology studies properties preserved under continuous deformation.

Topological Properties and Invariants

Definition2.4Topological Property

A property $P$ of topological spaces is a topological property (or topological invariant) if whenever $X$ has property $P$ and $Y \cong X$ , then $Y$ also has property $P$ .

Equivalently, $P$ is preserved by homeomorphisms.

ExampleExamples of Topological Properties

The following are topological properties:

Compactness, connectedness, path-connectedness
Hausdorff, regular, normal
First-countable, second-countable, separable
Metrizable
Having a given cardinality of open sets

The following are not topological properties:

Boundedness (depends on the metric, not the topology)
Completeness (depends on the metric; $(0,1) \cong \mathbb{R}$ but $(0,1)$ is not complete while $\mathbb{R}$ is)

Topological Embeddings

Definition2.5Topological Embedding

A function $f: X \to Y$ is a topological embedding if $f$ is an injective continuous map and $f: X \to f(X)$ is a homeomorphism, where $f(X)$ carries the subspace topology from $Y$ .

Equivalently, $f$ is an embedding if and only if $f$ is injective, continuous, and for every open $U \subseteq X$ , $f(U)$ is open in $f(X)$ (i.e., $f(U) = V \cap f(X)$ for some open $V \subseteq Y$ ).

ExampleEmbeddings

The inclusion $\iota: A \hookrightarrow X$ of a subspace is always an embedding.
The map $f: \mathbb{R} \to \mathbb{R}^2$ defined by $f(t) = (t, t^2)$ is an embedding (the image is a parabola, homeomorphic to $\mathbb{R}$ ).
The map $f: [0, 1] \to \mathbb{R}^2$ defined by $f(t) = (\cos 2\pi t, \sin 2\pi t)$ is not an embedding, since $f(0) = f(1)$ (not injective).

Local Homeomorphisms

Definition2.6Local Homeomorphism

A map $f: X \to Y$ is a local homeomorphism if every point $x \in X$ has an open neighborhood $U$ such that $f(U)$ is open in $Y$ and $f|_U: U \to f(U)$ is a homeomorphism.

ExampleCovering Maps as Local Homeomorphisms

The exponential map $\exp: \mathbb{R} \to S^1$ given by $t \mapsto e^{2\pi i t}$ is a local homeomorphism but not a global homeomorphism (it is not injective). For any $t_0 \in \mathbb{R}$ , the restriction to $(t_0 - 1/2, t_0 + 1/2)$ is a homeomorphism onto its image.

Every covering map is a local homeomorphism, but the converse is false.

RemarkHomeomorphism as an Equivalence Relation

The homeomorphism relation is an equivalence relation on topological spaces:

Reflexive: $\operatorname{id}: X \to X$ is a homeomorphism.
Symmetric: If $f: X \to Y$ is a homeomorphism, so is $f^{-1}: Y \to X$ .
Transitive: If $f: X \to Y$ and $g: Y \to Z$ are homeomorphisms, so is $g \circ f: X \to Z$ .

The equivalence classes are called homeomorphism types or topological types.