A locally compact non-k_2-space

spaces/S000211/README.md

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+---
+uid: S000211
+name: Isbell's locally compact non-k-space topology on $\omega_1+1$
+refs:
+  - doi: 10.1090/S0002-9939-1987-0891170-3
+    name: A distinguishing example in k-spaces (Isbell)
+  - mathse: 3904987
+    name: Answer to "Is there a locally compact space which is not a k-space"
+---
+
+The set $\omega_1+1$ with the topology generated by the basis
+$\{[0,\omega_1)\setminus F:F\text{ finite}\}\cup\{[\alpha,\omega_1]\setminus F:\alpha<\omega_1,F\text{ finite}\}$.
+
+Constructed by Isbell in
+{{doi:10.1090/S0002-9939-1987-0891170-3}}
+as an example of a {P130} space which is not {P141}.
+See also {{mathse:3904987}}.

spaces/S000211/properties/P000002.md

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+---
+space: S000211
+property: P000002
+value: true
+---
+
+By construction.

spaces/S000211/properties/P000039.md

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+---
+space: S000211
+property: P000039
+value: true
+---
+
+By construction.

spaces/S000211/properties/P000099.md

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+---
+space: S000211
+property: P000099
+value: false
+---
+
+The sequence $x_n=n$ converges to both $0$ and $1$.

spaces/S000211/properties/P000101.md

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+---
+space: S000211
+property: P000101
+value: false
+refs:
+  - mathse: 5017415
+    name: Answer to "Is there a locally compact space which is not a k-space"
+---
+
+The retract $[1,\omega_1]$ of $f:X\to X$ defined by $f(0)=1$ and $f(x)=x$ otherwise
+is not closed; see {{mathse:5017415}}.

spaces/S000211/properties/P000114.md

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+---
+space: S000211
+property: P000114
+value: true
+---
+
+By construction.

spaces/S000211/properties/P000141.md

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+---
+space: S000211
+property: P000141
+value: false
+refs:
+  - doi: 10.1090/S0002-9939-1987-0891170-3
+    name: A distinguishing example in k-spaces (Isbell)
+  - mathse: 5017415
+    name: Answer to "Is there a locally compact space which is not a k-space"
+---
+
+Shown in {{doi:10.1090/S0002-9939-1987-0891170-3}}.
+See {{mathse:5017415}} for details.

spaces/S000211/properties/P000191.md

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+---
+space: S000211
+property: P000191
+value: false
+---
+
+Any countable intersection of open sets is infinite.

spaces/S000211/properties/P000208.md

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+---
+space: S000211
+property: P000208
+value: true
+refs:
+  - doi: 10.1090/S0002-9939-1987-0891170-3
+    name: A distinguishing example in k-spaces (Isbell)
+---
+
+Given a cover of a non-empty subspace $S$, the least element of the subspace belongs to
+a member of the cover that contains $S\setminus F$ for some finite set $F$.