Homeomorphism

A homeomorphism is a map between two topological spaces that is bijective and continuous and whose inverse is also continuous. Spaces related by such a map are called homeomorphic: their points may be represented differently, but their topological structures agree. Homeomorphism is the fundamental notion of equivalence in topology, distinguishing properties of spaces from features of a particular geometric representation. (open.edu)

Definition and equivalent characterizations

Let $X$ and $Y$ be topological spaces. A function $f:X\to Y$ is a homeomorphism when:

1. It is a bijection, so every point of $Y$ corresponds to exactly one point of $X$.
2. It is a continuous function.
3. Its inverse function $f^{-1}:Y\to X$ is continuous.

Continuity means that the inverse image of every open set is open. Consequently, a bijection is a homeomorphism precisely when

Thus a homeomorphism transports the entire collection of open sets, not merely individual points. Equivalently, it is a continuous bijection that is an open map, or a continuous bijection that maps closed sets to closed sets. (math.ucla.edu)

The identity map is a homeomorphism, the inverse of a homeomorphism is a homeomorphism, and compositions of homeomorphisms are homeomorphisms. Hence being homeomorphic is an equivalence relation: it is reflexive, symmetric, and transitive. The notation $X\cong Y$ is often used when the intended equivalence is topological. (math.ucr.edu)

Examples and the inverse-continuity requirement

With their usual topologies, the interval $(0,1)$ and the positive real numbers are homeomorphic through

Likewise, $x\mapsto x^3$ is a homeomorphism from $\mathbb R$ to itself. These examples show that homeomorphisms need not preserve lengths or uniform rates of stretching. A circle and the boundary of a square are also homeomorphic, despite their different geometric appearances. (math.ucla.edu)

A continuous bijection alone is insufficient. Consider

where the interval and circle have their usual subspace topologies. This map is continuous and bijective, but its inverse is discontinuous at $(1,0)$. Points approaching that point around the circle from one direction have inverse images approaching $2\pi$, rather than $0$. The map joins the two ends topologically without supplying a continuous way to undo that identification. (jde27.uk)

Homeomorphism also does not depend on the dimension of the surrounding Euclidean space. The standard torus in $\mathbb R^3$ and the set

are both homeomorphic to $S^1\times S^1$, equipped with its product topology. Their intrinsic topology agrees although their ambient representations differ. (jde27.uk)

A compactness criterion

An important criterion states that every continuous bijection from a compact space to a Hausdorff space is a homeomorphism. The inverse therefore need not be checked separately under these hypotheses. (math.ucla.edu)

The proof uses closed sets. If $C\subseteq X$ is closed and $X$ is compact, then $C$ is compact. Its continuous image $f(C)$ is compact, and compact subsets of a Hausdorff space are closed. Thus $f$ maps closed sets to closed sets, which makes its inverse continuous. This criterion frequently establishes equivalence between a parametrized space and its geometric image. (math.ucla.edu)

Topological invariants

A topological invariant is a property or associated object unchanged by homeomorphism. Examples include compactness, connectedness, path connectedness, and the Hausdorff property. Such invariants can show that no homeomorphism exists. For instance, a compact space cannot be homeomorphic to a noncompact one. (pi.math.cornell.edu)

Point removal provides another useful test. Removing an interior point from $[0,1]$ disconnects it, whereas removing any single point from the square $[0,1]^2$ leaves a connected space. A homeomorphism would restrict to a homeomorphism between the corresponding punctured spaces, so the interval and square cannot be homeomorphic. This argument also explains why a continuous map filling the square cannot simultaneously be injective. (pi.math.cornell.edu)

In algebraic topology, homeomorphisms induce isomorphisms between associated algebraic objects, including homology groups. Different invariants obstruct homeomorphism, although agreement of selected invariants does not itself construct a homeomorphism. (math.uchicago.edu)

Related notions

A homotopy equivalence is weaker: its maps need only compose to maps homotopic to the identities, rather than exactly to the identities. Every homeomorphism is a homotopy equivalence, but a disk and a single point are homotopy equivalent without being homeomorphic. Similarly, the punctured plane is homotopy equivalent to a circle, although one is noncompact and the other compact. (math.uchicago.edu)

A diffeomorphism imposes smoothness on both a map and its inverse. The homeomorphism $x\mapsto x^3$ is not a diffeomorphism with the usual smooth structures because its inverse is not differentiable at zero. (people.math.harvard.edu)

A local homeomorphism has the homeomorphism property on suitable neighborhoods, but need not be globally bijective. The map $t\mapsto(\cos t,\sin t)$ from $\mathbb R$ to the circle is an example. Local homeomorphisms also underlie the definition of a manifold, whose points have neighborhoods homeomorphic to Euclidean space. (math.ucla.edu)