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nLab > Latest Changes: space of finite subsets

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeOct 17th 2019
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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>

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v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeOct 17th 2019
- (edited Oct 17th 2019)
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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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ be an non-empty regular topological space and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \geq 2 \in \mathbb{N}</annotation></semantics></math>$ .

Then the injection
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><msup><mi>exp</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>exp</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Conf_n(X) \hookrightarrow \exp^n(X)/\exp^{n-1}(X) </annotation></semantics></math>$
of the unordered configuration space of n points of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ into the quotient space of the space of finite subsets of cardinality $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\leq n</annotation></semantics></math>$ by its subspace of subsets of cardinality $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\leq n-1</annotation></semantics></math>$ is an open subspace-inclusion.

Moreover, if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ is compact, then so is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>exp</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>exp</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp^n(X)/\exp^{n-1}(X)</annotation></semantics></math>$ and the inclusion exhibits the one-point compactification $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mo lspace="0.11111em" rspace="0em">+</mo></msup></mrow><annotation encoding="application/x-tex">\big( Conf_n(X) \big)^{+} </annotation></semantics></math>$ of the configuration space:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mo lspace="0.11111em" rspace="0em">+</mo></msup><mspace width="0.27778em"/><mo>≃</mo><mspace width="0.27778em"/><msup><mi>exp</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>exp</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> \big( Conf_n(X) \big)^{+} \;\simeq\; \exp^n(X)/\exp^{n-1}(X) \,. </annotation></semantics></math>$
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nLab > Latest Changes: space of finite subsets