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
Every CW-complex is a paracompactum, i.e., a paracompact Hausdorff space.
For one textbook explanation, see e.g. Fritsch-Piccinini 90, theorem 1.3.5. Below we give a somewhat more categorically minded proof, linking to relevant results elsewhere in the nLab.
Let $X_n$ denote the $n^{th}$ skeleton of $X$. We argue by induction that each skeleton is a paracompactum. Vacuously $X_{-1} = \emptyset$ is a paracompactum. Now suppose $X_{n-1}$ is a paracompactum, and suppose $X_n$ is formed as an attachment space with attaching map $f: \sum_{i \in I} S_i^{n-1} \to X_{n-1}$, so that
is a pushout square. The embedding $h$ is a closed embedding, and so its pushout along $f$ is a closed embedding $k$. Furthermore, the spheres $S_i^{n-1}$ and disks $D_i^n$ are paracompacta since they are compact Hausdorff, and coproducts of paracompacta are again paracompacta, making $h$ a closed embedding of paracompact Hausdorff spaces. By the result on pushouts of closed embeddings of paracompacta, it now follows that $X_n$ is a paracompactum.
Thus the CW-complex $X$ is a colimit of a sequence of closed embeddings
between paracompacta. It follows that this colimit is a paracompactum.
maps from compact spaces to Hausdorff spaces are closed and proper
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
That every CW complexes is Hausdorff (in fact normal) may be folklore, a proof is spelled out in:
An early original article with the paracompactness statement is:
Textbook account:
Last revised on November 21, 2021 at 07:24:03. See the history of this page for a list of all contributions to it.