see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
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
locally compact and second-countable spaces are sigma-compact
Theorems
Basic homotopy theory
Serre fibration $\Leftarrow$ Hurewicz fibration $\Rightarrow$ Dold fibration $\Leftarrow$ shrinkable map
Write
for the set of inclusions of the topological n-disks, into their cylinder objects, along (for definiteness) the left endpoint inclusion.
A Serre fibration is a $J_{Top}$-injective morphism, def. 1, hence a continuous function $f \colon X \longrightarrow Y$ that has the right lifting property with respect to all inclusions of the form $(id,0) \colon D^n \hookrightarrow D^n \times I$ that include the standard topological n-disk into its standard cylinder object.
I.e. $f$ is a Serre fibration if for every commuting square of continuous functions of the form
then there exists a continuous function $h \colon D^n \to X$ such as to make a commuting diagram of the form
The condition in def. 2 is part of the condition on a Hurewicz fibration, hence every Hurewicz fibration is in particular a Serre fibration.
The class of Serre fibrations serves as the class of abstract fibrations in the classical model structure on topological spaces (whence “Serre-Quillen model structure”).
A Serre fibration has the right lifting property against all retracts of $J_{Top}$-relative cell complexes (def. 1).
By general closure properties of projective and injective morphisms, see there this proposition for details.
Since Serre fibrations are the abstract fibrations in the Serre-classical model structure on topological spaces, the following statement follows from general model category theory. But it may also be seen by direct inspection, as follows.
For $X$ a finite CW-complex, then its inclusion $X \overset{(id, \delta_0)}{\longrightarrow} X \times I$ into its standard cylinder is a $J_{Top}$-relative cell complex (def. 1).
First erect a cylinder over all 0-cells
Assume then that the cylinder over all $n$-cells of $X$ has been erected using attachment from $J_{Top}$. Then the union of any $(n+1)$-cell $\sigma$ of $X$ with the cylinder over its boundary is homeomorphic to $D^{n+1}$ and is like the cylinder over the cell “with end and interior removed”. Hence via attaching along $D^{n+1} \to D^{n+1}\times I$ the cylinder over $\sigma$ is erected.
Let $f\colon X \longrightarrow Y$ be a Serre fibration, def. 2, let $y \colon \ast \to Y$ be any point and write
for the fiber inclusion over that point. Then for every choice $x \colon \ast \to X$ of lift of the point $y$ through $f$, the induced sequence of homotopy groups
is exact, in that the kernel of $f_\ast$ is canonically identified with the image of $\iota_\ast$:
It is clear that the image of $\iota_\ast$ is in the kernel of $f_\ast$ (every sphere in $F_y\hookrightarrow X$ becomes constant on $y$, hence contractible, when sent forward to $Y$).
For the converse, let $[\alpha]\in \pi_{\bullet}(X,x)$ be represented by some $\alpha \colon S^{n-1} \to X$. Assume that $[\alpha]$ is in the kernel of $f_\ast$. This means equivalently that $\alpha$ fits into a commuting diagram of the form
where $\kappa$ is the contracting homotopy witnessing that $f_\ast[\alpha] = 0$.
Now since $x$ is a lift of $y$, there exists a left homotopy
as follows:
(for instance: regard $D^n$ as embedded in $\mathbb{R}^n$ such that $0 \in \mathbb{R}^n$ is identified with the basepoint on the boundary of $D^n$ and set $\eta(\vec v,t) \coloneqq \kappa(t \vec v)$).
The pasting of the top two squares that have appeared this way is equivalent to the following commuting square
Because $f$ is a Serre fibration and by lemma 1 and prop. 1, this has a lift
Notice that $\tilde \eta$ is a basepoint preserving left homotopy from $\alpha = \tilde \eta|_1$ to some $\alpha' \coloneqq \tilde \eta|_0$. Being homotopic, they represent the same element of $\pi_{n-1}(X,x)$:
But the new representative $\alpha'$ has the special property that its image in $Y$ is not just trivializable, but trivialized: combining $\tilde \eta$ with the previous diagram shows that it sits in the following commuting diagram
The commutativity of the outer square says that $f_\ast \alpha'$ is constant, hence that $\alpha'$ is entirely contained in the fiber $F_y$. Said more abstractly, the universal property of fibers gives that $\alpha'$ factors through $F_y\overset{\iota}{\hookrightarrow} X$, hence that $[\alpha'] = [\alpha]$ is in the image of $\iota_\ast$.
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long exact sequence of homotopy groups
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