topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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
continuous metric space valued function on compact metric space is uniformly continuous
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
Analysis Theorems
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Given a sequence
of (pointed) topological spaces, then its mapping telescope is the result of forming the (reduced) mapping cylinder $Cyl(f_n)$ for each $n$ and then attaching all these cylinders to each other in the canonical way.
At least if all the $f_n$ are inclusions, this is the sequential attachment of ever “larger” cylinders, whence the name “telescope”.
The mapping telescope is a representation for the homotopy colimit over $X_\bullet$. It is used for instance for discussion of lim^1 and Milnor sequences (and that’s maybe the origin of the concept?).
For
a sequence in Top, its mapping telescope is the quotient topological space of the disjoint union of product topological spaces
where the equivalence relation quotiented out is
for all $n\in \mathbb{N}$ and $x_n \in X_n$.
Analogously for $X_\bullet$ a sequence of pointed topological spaces then use reduced cylinders to set
For $X_\bullet$ the sequence of stages of a (pointed) CW-complex $X = \underset{\longleftarrow}{\lim}_n X_n$, then the canonical map
from the mapping telescope, def. 1, is a weak homotopy equivalence.
Write in the following $Tel(X)$ for $Tel(X_\bullet)$ and write $Tel(X_n)$ for the mapping telescop of the substages of the finite stage $X_n$ of $X$. It is intuitively clear that each of the projections at finite stage
is a homotopy equivalence, hence a weak homotopy equivalence. A concrete construction of a homotopy inverse is given for instance in (Switzer 75, proof of prop. 7.53).
Moreover, since spheres are compact, so that elements of homotopy groups $\pi_q(Tel(X))$ are represented at some finite stage $\pi_q(Tel(X_n))$ it follows that
are isomorphisms for all $q\in \mathbb{N}$ and all choices of basepoints (not shown).
Together these two facts imply that in the following commuting square, three morphisms are isomorphisms, as shown.
Therefore also the remaining morphism is an isomorphism (two-out-of-three). Since this holds for all $q$ and all basepoints, it is a weak homotopy equivalence.
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 $\,$ |