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Add results to show that pseudometrizable implies 1st countable #244

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Merged
ccaruvana merged 4 commits into from
Mar 16, 2023
Merged

Add results to show that pseudometrizable implies 1st countable #244

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0f1a0ff
Add results to show that pseudometrizable implies 1st countable
StevenClontz Mar 15, 2023
c59a23b
fix T327 id
StevenClontz Mar 15, 2023
1672e64
fix typos in P144
StevenClontz Mar 15, 2023
7ca0fea
add mathse reference
StevenClontz Mar 16, 2023
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10 changes: 10 additions & 0 deletions properties/P000144.md
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Original file line number Diff line number Diff line change
@@ -0,0 +1,10 @@
---
uid: P000144
slug: locally-pseudometrizable
name: Locally pseudometrizable
refs:
- doi: 10.1090/S0002-9947-1969-0251719-4
name: TODO
---

Every point has a neighborhood that is {P121}.
10 changes: 4 additions & 6 deletions theorems/T000057.md
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@@ -1,14 +1,12 @@
---
uid: T000057
if:
P000121: true
P000144: true
then:
P000028: true
refs:
- doi: 10.1007/978-1-4612-6290-9
name: Counterexamples in Topology
- mathse: 4659902
name: How can the implication metrizable -> first countable be generalized?
---

If $X$ is pseudometric, for any $x \in X$, $\{B(x,\frac{1}{n})\,|\,n \in \omega\}$ is a countable local base at $x$.

Asserted on page 34-35 of {{doi:10.1007/978-1-4612-6290-9}} for metric spaces. The same holds for pseudometric spaces.
Take a pseduometrizable neighborhood. The balls of rational radius within this neighborhood form a local basis.
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@ccaruvana ccaruvana Mar 16, 2023 •
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I didn't catch this before accepting the pull request, but "pseudometrizable" is misspelled; it appears as "pseduometrizable"

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@StevenClontz StevenClontz Mar 16, 2023

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I'll make a hotfix for this directly on main.

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@prabau prabau Mar 16, 2023

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I'll make a hotfix for this directly on main.

Seems T327 has the same missplelling.

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@StevenClontz StevenClontz Mar 16, 2023

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Fixed now in 70e67cd. I'm okay if maintainers/reviewers do trivial (literally one line) typo changes directly on main.

Go to e.g. https://github.com/pi-base/data/blob/main/theorems/T000327.md click the pencil icon to get a quick editor, make the fix, change the commit message to say "fixed typo" (I forgot to do this at 70e67cd), then commit directly to main.

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@prabau prabau Mar 16, 2023

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Not sure it would let me do this. Just to check, I clicked on the pencil icon and at the bottom it says "you can't commit to main, because it's a protected branch. Create a new branch for this commit and start a pull request."

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@StevenClontz StevenClontz Mar 17, 2023

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Oops. I guess that might be an admin-only feature. So in that case I still recommend the workflow when convenient, but rather than committing to main, create a "patch" branch and open a PR that can be quickly reviewed/accepted by others.

12 changes: 12 additions & 0 deletions theorems/T000326.md
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Original file line number Diff line number Diff line change
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---
uid: T000326
if:
P000121: true
then:
P000144: true
refs:
- mathse: 4659902
name: How can the implication metrizable -> first countable be generalized?
---

The entire space is the desired neighborhood.
12 changes: 12 additions & 0 deletions theorems/T000327.md
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---
uid: T000327
if:
P000082: true
then:
P000144: true
refs:
- mathse: 4659902
name: How can the implication metrizable -> first countable be generalized?
---

Every metric is a pseduometric.