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S136 is scattered #1717

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S136 is scattered #1717

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1e970e1
s136 scattered
felixpernegger Mar 30, 2026
96839c9
wrong uid
felixpernegger Mar 30, 2026
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3 changes: 0 additions & 3 deletions spaces/S000136/properties/P000019.md
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Original file line number Diff line number Diff line change
Expand Up @@ -2,9 +2,6 @@
space: S000136
property: P000019
value: false
refs:
- zb: "0386.54001"
name: Counterexamples in Topology
---

Closed subspaces of {P19} spaces are {P19}, but the closed subspace {S137} of this space is not {P19}.
9 changes: 9 additions & 0 deletions spaces/S000136/properties/P000051.md
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@mathmaster13 mathmaster13 Apr 4, 2026

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Hi! I tried to review this but got stuck figuring out what it meant. It looks to me like you're integrating a function $\chi_{{r}}$ with a set $M$ and getting another function $x_r$ back? What does that mean?

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@mathmaster13 mathmaster13 Apr 4, 2026

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The other 3 files are tiny and look okay to me.

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@felixpernegger felixpernegger Apr 4, 2026

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Hi! I tried to review this but got stuck figuring out what it meant. It looks to me like you're integrating a function χ r with a set M and getting another function x r back? What does that mean?

The $x_r$ are in the definition of the space, as well as $M$.
However I do realise that the way I wrote the argument does not actually make sense (the proof in my head does, but i wrote it down super wrongly), so thank you, I'll rewrite it.

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---
space: S000136
property: P000051
value: true
---

For $r \in \mathbb{R}$ consider the characteristic function $\chi_{\{r\}}:2^\mathbb{R}\to 2$. Then $\chi_{\{r\}} \cap M = x_r$, therefore $M$ is discrete as a subspace.

Thus if $A \subseteq X$ contains no elements of $F$ we are done, otherwise $A$ already contains an isolated point by definition.
10 changes: 0 additions & 10 deletions spaces/S000136/properties/P000139.md
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7 changes: 7 additions & 0 deletions spaces/S000137/properties/P000051.md
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Original file line number Diff line number Diff line change
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---
space: S000137
property: P000051
value: true
---

$X$ is a subspace of {S136} and {S136|P51}.
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