Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological

Let $(X,{\mathcal{T}}_X)$ and $(Y,{\mathcal{T}}_Y)$ be topological spaces and let $f:X\to Y$ be a function. Continuity is as in Continuity of a map of topological spaces at a point and globally, injections, surjections and bijections as in Injection, surjection, bijection.

• $f$ is an open map if $f[U]$ is open in $Y$ for every open $U\subseteq X$.
• $f$ is a closed map if $f[F]$ is closed in $Y$ for every closed $F\subseteq X$.
• $f$ is a homeomorphism if $f$ is a continuous bijection whose inverse $f^{-1}:Y\to X$ is also continuous. The spaces are homeomorphic, written $X\cong Y$, when a homeomorphism $X\to Y$ exists.
• $f$ is an embedding if $f$ is injective and the corestriction $f_0:X\to f[X]$, $f_0(x)=f(x)$, is a homeomorphism onto $f[X]$ carrying the subspace topology inherited from $Y$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

The inverse in the third clause exists because $f$ is a bijection, and it is the unique two-sided inverse (Injection, surjection, bijection); no choice principle is involved. Continuity of $f^{-1}$ is a genuine additional demand: a continuous bijection need not be a homeomorphism, and this page records that failure as a false statement with a two-point witness.

Open, closed and homeomorphism are three different conditions. A homeomorphism is continuous by definition, but an open map need not be continuous and a closed map need not be continuous; and continuity implies neither openness nor closedness. An open map need not be closed and a closed map need not be open, and Sierpinski space $S=\{a,b\}$ with open point $b$ (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies) witnesses both failures at once: the constant map $S\to S$ with value $b$ is open, since the image of every nonempty set is the open set $\{b\}$, and is not closed, since the image of the closed set $\{a\}$ is $\{b\}$, whose complement $\{a\}$ is not open; the constant map with value $a$ is closed and not open by the same computation read the other way. What is true is that for a continuous bijection the three notions collapse: it is a homeomorphism exactly when it is open, exactly when it is closed. That is proved in the next item and is not assumed here.

Topological properties. A property of topological spaces is a condition $P$ that is either true or false of each space. $P$ is a topological property (one is also said to be preserved by homeomorphism, or invariant) when $X\cong Y$ implies that $P(X)$ and $P(Y)$ have the same truth value. Since $\cong$ is an equivalence relation on spaces — the identity is a homeomorphism, inverses and composites of homeomorphisms are homeomorphisms, all three verified in the next item — a topological property is exactly one that is constant on each $\cong$-class.

What a homeomorphism transports. If $h:X\to Y$ is a homeomorphism then $U\mapsto h[U]$ is a bijection from ${\mathcal{T}}_X$ onto ${\mathcal{T}}_Y$, with inverse $V\mapsto h^{-1}[V]$: both maps are well defined because $h$ and $h^{-1}$ are continuous, and they are mutually inverse because $h$ is a bijection. So a homeomorphism is an isomorphism of the structure "a set together with a distinguished family of subsets", and every notion defined from the open sets alone — closed, closure, interior, boundary, dense, convergence of sequences, continuity of maps into and out of the space — is carried across by it. Anything defined from extra data, such as a metric or an order, is not, and that distinction is exactly what the phrase topological property is for.

• Being an embedding is not the same as being injective and continuous. The identity from a set with the discrete topology to the same set with a coarser topology is injective and continuous, its image is the whole space, and it is an embedding only if the two topologies agree. The extra content of "embedding" is that the source topology is recovered as the trace of the target one, which is precisely the characteristic property of the subspace topology.

• The inclusion of a subspace is the model embedding. For $S\subseteq X$ with the subspace topology, $\iota :S\to X$ is injective and its corestriction to $\iota [S]=S$ is the identity of $(S,{\mathcal{T}}_S)$, hence a homeomorphism; so $\iota$ is an embedding (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace). Conversely an embedding $f:X\to Y$ identifies $X$ with the subspace $f[X]$ of $Y$, which is the sense in which "$X$ sits inside $Y$" is ever asserted in this library.

• The notation $X\cong Y$ hides the map, and sometimes that matters. Two spaces may be homeomorphic by many different homeomorphisms, and no canonical one is claimed by the notation. Where a specific map is used it is named.