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
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
locally compact and second-countable spaces are sigma-compact
Theorems
Basic homotopy theory
Serre fibration $\Leftarrow$ Hurewicz fibration $\Rightarrow$ Dold fibration$\Leftarrow$ shrinkable map
A Dold fibration is a continuous map of topological spaces that allows lifting of homotopies, with initial condition agreeing with a given map up to a vertical homotopy.
The morphism $f:E \to B$ of topological spaces is said to have the weak covering homotopy property (WCHP) for the space $Y$ if for all squares
there is a map $\hat{g}:Y\times I \to E$ such that $f\circ \hat{g} = g$ and there is a vertical homotopy between $\hat{g}(-,0):Y\to E$ and $g_0$. The synonymous expression weak homotopy lifting property (WHLP) is also used.
A continuous map is a Dold fibration if it has the WCHP for all spaces. Somewhat surprisingly, there is an equivalent condition in terms of delayed homotopies. A delayed homotopy is a homotopy $H:Y\times I\to Z$ such that $H(y,t)=H(y,0)$ for $0\leq t\leq t_0$ for some $t_0\gt 0$. A continuous map is a Dold fibration iff in the diagram above in which $g$ is a delayed homotopy, can be filled with a diagonal map $\hat{g}:Y\times I \to E$ such that the diagram is strictly commutative. It is of course not required that $\hat{g}$ be delayed (one can require, but then one allows $t_0$ for $\hat{g}$ to be possibly smaller than $t_0$ for $g$). This is sometimes called the delayed homotopy lifting property.
Every Hurewicz fibration is a Dold fibration.
Not all Serre fibrations are Dold fibrations.
David Roberts: Can we come up with a counterexample of a Serre fibration that isn’t a Dold fibration? I’ll ask on MathOverflow.
Not all Dold fibrations are Serre fibrations.
Here is a very simple counter-example due to Dold. Consider the union of line segments
in $\mathbf{R}^2$, and the map projecting on to the first coordinate, $pr_1:E \to [-1,1]$. Then this map is a Dold fibration but not a Serre fibration.
Chris Schommer-Pries: I believe this is an example of a quasi-fibration which is not a Serre fibration, but it is not a Dold fibration either.
One could consider maps that have the WCHP for just cubes – these are a sort of hybrid Dold–Serre fibration (warning! nonstandard terminology. I just made it up, suggestions appreciated). For these maps there exists a long exact sequence in homotopy once basepoints are chosen. For classes of maps determined by (homotopy) lifting properties, this is about the minimum one needs to define such a long exact sequence. On the other hand, quasifibrations give rise to a long exact sequence in homotopy, but are defined by homotopy properties of the fibres.
Every shrinkable map is a Dold fibration.
This result follows from a theorem of Dold about locally homotopy trivial map?s being the same as Dold fibrations. It should be obvious that a shrinkable map is globally homotopy trivial, with trivial fibre.
Let $\{U_i \to X\}$ be a numerable open cover. Let $C(\{U_i\})$ the Cech nerve (a simplicial object in topological spaces) and $|C(\{U_i\})| \in Top$ its geometric realization. Then the canonical map
is shrinkable, hence a Dold fibration.
This observation is due to Segal. See shrinkable map.