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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 17th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexPage created, but author did not leave any comments. <a href="https://ncatlab.org/nlab/revision/space+of+finite+subsets/1">v1</a>, <a href="https://ncatlab.org/nlab/show/space+of+finite+subsets">current</a>

   Page created, but author did not leave any comments.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 17th 2019
   • (edited Oct 17th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting something, for the moment just so as to record this fact: *** Let $X$ be an [[inhabited set|non-empty]] [[regular topological space]] and $n \geq 2 \in \mathbb{N}$. Then the injection \[ Conf_n(X) \hookrightarrow \exp^n(X)/\exp^{n-1}(X) \] 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 topological space|compact]], then so is $\exp^n(X)/\exp^{n-1}(X)$ and the inclusion exhibits the [[one-point compactification]] $\big( Conf_n(X) \big)^{+} $ of the configuration space: $$ \big( Conf_n(X) \big)^{+} \;\simeq\; \exp^n(X)/\exp^{n-1}(X) \,. $$ <a href="https://ncatlab.org/nlab/revision/space+of+finite+subsets/1">v1</a>, <a href="https://ncatlab.org/nlab/show/space+of+finite+subsets">current</a>

   starting something, for the moment just so as to record this fact:

   ---

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

   Then the injection

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

   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 exhibits the one-point compactification $\big( Conf_n(X) \big)^{+}$ of the configuration space:

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

   v1, current