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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 17th 2019

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• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 17th 2019
• (edited Oct 17th 2019)

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:

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