nLab space of finite subsets

Idea

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).

Properties

Relation to unordered configuration space of points

The topological subspace of finite subsets of cardinality exactly equal to $n$ is the unordered configuration space of points in $X$

(1)$$Conf_n(X) \hookrightarrow \exp^n(X)$$

Proposition

Let $X$ be an non-empty regular topological space and $n \geq 2 \in \mathbb{N}$.

Then the injection (1)

(2)$$Conf_n(X) \hookrightarrow \exp^n(X)/\exp^{n-1}(X)$$

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:

$$\big( Conf_n(X) \big)^{+} \;\simeq\; \exp^n(X)/\exp^{n-1}(X) \,.$$

(Handel 00, Prop. 2.23, see also Félix-Tanré 10)

References

• 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)