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
For $X$ a topological space and $n \in \mathbb{N}$ a natural number, the space of finite subsets of cardinality $\leq n$ in $X$ is the suitably topologized set of finite subsets $S \subset X$ of cardinality $\left\vert S\right\vert \leq n$, often denoted $\exp^n X$ or similar (e.g. Félix-Tanré 10).
The topological subspace of finite subsets of cardinality exactly equal to $n$ is the unordered configuration space of points in $X$
Let $X$ be an non-empty regular topological space and $n \geq 2 \in \mathbb{N}$.
Then the injection (1)
of the unordered configuration space of n points of $X$ into the quotient space of the space of finite subsets of cardinality $\leq n$ by its subspace of subsets of cardinality $\leq n-1$ is an open subspace-inclusion.
Moreover, if $X$ is compact, then so is $\exp^n(X)/\exp^{n-1}(X)$ and the inclusion (2) exhibits the one-point compactification $\big( Conf_n(X) \big)^{+}$ of the configuration space:
(Handel 00, Prop. 2.23, see also Félix-Tanré 10)
David Handel, Some Homotopy Properties of Spaces of Finite Subsets of Topological Spaces, Houston Journal of Mathematics, Electronic Edition Vol. 26, No. 4, 2000 (pdfhjm:Vol26-4)
Yves Félix, Daniel Tanré Rational homotopy of symmetric products and Spaces of finite subsets, Contemp. Math 519 (2010): 77-92 (pdf)
Jacob Mostovoy, Rustam Sadykov, On the connectivity of finite subset spaces (arXiv:1203.5180)
Last revised on October 17, 2019 at 10:45:27. See the history of this page for a list of all contributions to it.