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
In algebraic topology by a CW-pair $(X,A)$ is meant a CW-complex $X$ equipped with a sub-complex inclusion $A \hookrightarrow X$.
The concept appears prominently in the discussion of ordinary relative homology and generally in the Eilenberg-Steenrod axioms for generalized homology/generalized cohomology.
For $X$ a CW complex, the inclusion $A \hookrightarrow X$ of any subcomplex has an open neighbourhood in $X$ which is a deformation retract of $A$. In particular such an inclusion is a good pair in the sense of relative homology.
For instance (Hatcher, prop. A.5).
For $(X,A)$ a CW-pair, then the $A$-relative singular homology of $X$ coincides with the reduced singular homology of the quotient space $X/A$:
For instance (Hatcher, prop. 2.22).
By assumption we can find a neighbourhood $A \stackrel{j}{\to} U \hookrightarrow X$ such that $A \hookrightarrow U$ has a deformation retract and hence in particular is a homotopy equivalence and so induces also isomorphisms on all singular homology groups.
It follows in particular that for all $n \in \mathbb{N}$ the canonical morphism $H_n(X,A) \stackrel{H_n(id,j)}{\to} H_n(X,U)$ is an isomorphism, by homotopy invariance of relative singular homology.
Given such $U$ we have an evident commuting diagram of pairs of topological spaces
Here the right vertical morphism is in fact a homeomorphism.
Applying relative singular homology to this diagram yields for each $n \in \mathbb{N}$ the commuting diagram of abelian groups
Here the left horizontal morphisms are the above isomorphims induced from the deformation retract. The right horizontal morphisms are isomorphisms by excision and the right vertical morphism is an isomorphism since it is induced by a homeomorphism. Hence the left vertical morphism is an isomorphism (2-out-of-3 for isomorphisms).
Allen Hatcher, Algebraic Topology, 2002
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 5.1 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Last revised on November 23, 2021 at 21:30:02. See the history of this page for a list of all contributions to it.