see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
fiber space, space attachment
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
subsets are closed in a closed subspace precisely if they are closed in the ambient space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
Theorems
Given a topological space $X$ in the sense of (Bourbaki 71) (that is, a set $X$ and a topology $\tau_X$) and a subset $Y$ of $X$, a topology $\tau_Y$ on $Y$ is said to be the topology induced from $\tau_X$ by the set inclusion $Y \hookrightarrow X$ if $\tau_Y = \tau_X \cap_{pw} \{Y\} \coloneqq \{ U \cap Y | U\in\tau_X\}$. In other words, $\tau_Y$ is the smallest topology on $Y$ such that the inclusion $Y \hookrightarrow X$ is continuous (the initial topology on that map). The pair $(Y,\tau_Y)$ is then said to be a topological subspace of $(X,\tau_X)$. The induced topology is for that reason sometimes called the subspace topology on $Y$.
A continuous function that factors as a homeomorphism onto its image equipped with the subspace topology is called an embedding of topological spaces.
A property of topological spaces is said to be hereditary if its satisfaction for a topological space $X$ implies its satisfaction for all topological subspaces of $X$.
The open subsets of a closed interval $[0,1] \subset \mathbb{R}$ regarded as a topological subspace of the real line equipped with its Euclidean metric topology is generated from the sub-base $\beta = \{ [0, a) ,(a,1]\}_{a \in [0,1]}$.
The image on the right shows open subsets in the closed square $[0,1]^2$, regarded as a topological subspace of the Euclidean plane
A subspace $i: Y \hookrightarrow X$ is closed if $Y$ is closed as a subset of $X$ (or if $i$ is a closed map), and is open if $Y$ is open as a subset of $X$ (or if $i$ is an open map).
Topological subspace inclusions (topological embedding) are precisely the regular monomorphisms in the category Top of all topological spaces.
For example, the equalizer of two maps $f, g \colon X \stackrel{\to}{\to} Y$ in Top is computed as the equalizer at the underlying-set level, equipped with the subspace topology.
The pushout in Top of any (closed/open) subspace $i \colon A \hookrightarrow B$ along any continuous function $f \colon A \to C$ is a (closed/open) subspace $j: C \hookrightarrow D$.
Since $U = \hom(1, -): Top \to Set$ is faithful, we have that monos are reflected by $U$; also monos and pushouts are preserved by $U$ since $U$ has both a left adjoint and a right adjoint. In $Set$, the pushout of a mono along any map is a mono, so we conclude $j$ is monic in $Top$. Furthermore, such a pushout diagram in $Set$ is also a pullback, so that we have the Beck-Chevalley equality $\exists_i \circ f^\ast = g^\ast \exists_j \colon P(C) \to P(B)$ (where $\exists_i \colon P(A) \to P(B)$ is the direct image map between power sets, and $f^\ast: P(C) \to P(A)$ is the inverse image map).
To prove that $j$ is a subspace, let $U \subseteq C$ be any open set. Then there exists open $V \subseteq B$ such that $i^\ast(V) = f^\ast(U)$ because $i$ is a subspace inclusion. If $\chi_U \colon C \to \mathbf{2}$ and $\chi_V \colon B \to \mathbf{2}$ are the maps to Sierpinski space that classify these open sets, then by the universal property of the pushout, there exists a unique continuous map $\chi_W \colon D \to \mathbf{2}$ which extends the pair of maps $\chi_U, \chi_V$. It follows that $j^{-1}(W) = U$, so that $j$ is a subspace inclusion.
If moreover $i$ is an open inclusion, then for any open $U \subseteq C$ we have that $j^\ast(\exists_j(U)) = U$ (since $j$ is monic) and (by Beck-Chevalley) $g^\ast(\exists_j(U)) = \exists_i(f^\ast(U))$ is open in $B$. By the definition of the topology on $D$, it follows that $\exists_j(U)$ is open, so that $j$ is an open inclusion. The same proof, replacing the word “open” with the word “closed” throughout, shows that the pushout of a closed inclusion $i$ is a closed inclusion $j$.
A similar (but even simpler) line of argument establishes the following result.
Let $\kappa$ be an ordinal, viewed as a preorder category, and let $F: \kappa \to Top$ be a functor that preserves directed colimits. Then if $F(i \leq j)$ is a (closed/open) subspace inclusion for each morphism $i \leq j$ of $\kappa$, then the canonical map $F(0) \to colim_{i \in \kappa} F(i)$ is also a (closed/open) inclusion.
There is also a notion of a Grothendieck topology induced along a functor from a Grothendieck topology on another category (actually the input can be a somewhat more general coverage, then the topology induced along the identity functor will serve as a sort of a completion). (this will be explained later).
A topology may be induced by more than a function other than a subset inclusion, or indeed by a family of functions out of $Y$ (not necessarily all with the same target). However, the term ‘induced topology’ is often (usually?) restricted to subspaces; the general concept is called a weak topology. (This construction can be done in any topological concrete category; in this generality it is often called an initial structure for a source.) The dual construction (involving functions to $Y$) is a strong topology (or final structure for a sink); an example is the quotient topology on a quotient space.
examples of universal constructions of topological spaces:
$\phantom{AAAA}$limits | $\phantom{AAAA}$colimits |
---|---|
$\,$ point space$\,$ | $\,$ empty space $\,$ |
$\,$ product topological space $\,$ | $\,$ disjoint union topological space $\,$ |
$\,$ topological subspace $\,$ | $\,$ quotient topological space $\,$ |
$\,$ fiber space $\,$ | $\,$ space attachment $\,$ |
$\,$ mapping cocylinder, mapping cocone $\,$ | $\,$ mapping cylinder, mapping cone, mapping telescope $\,$ |
$\,$ cell complex, CW-complex $\,$ |