Add S51 Khalimsky line, an example of a non-T1 cut point space (#1167)

properties/P000193.md

@@ -14,4 +14,4 @@ refs:
 
 A space in which every open cover admits a shrinking; that is, a space $X$ in which, given any open cover $\{ U_\alpha : \alpha \in A\}$, there is an open cover $\{ V_\alpha : \alpha \in A\}$ such that $\overline{V_\alpha} \subseteq U_\alpha$ for each $\alpha \in A$.
 
-See also [Dan Ma's Topology Blog post on Spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).
+See also [Dan Ma's Topology Blog post on Spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).

spaces/S000051/README.md

@@ -0,0 +1,18 @@
+---
+uid: S000051
+name: Khalimsky line
+aliases:
+- Digital line
+refs:
+- doi: 10.1090/s0002-9939-99-04839-x
+  name: Cut-point spaces (Honari & Bahrampour)
+- doi: 10.32219/isms.80.1_15
+  name: On Generalized Digital Lines (Nakaoka, Tamari, Maki) 
+---
+
+The integers $\mathbb Z$ with $\{\{2i-1,2i,2i+1\}\mid i\in\mathbb Z\}\cup\{\{2i+1\}\mid i\in\mathbb Z\}$ as a basis for the topology.
+
+It is the unique {P205} such that no proper subspace is also a cut point space.
+See Theorem 4.5 of {{doi:10.1090/s0002-9939-99-04839-x}} for a proof.
+
+Also named the *digital line* in the subject of digital topology, e.g. in {{doi:10.32219/isms.80.1_15}}.

spaces/S000051/properties/P000010.md

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+---
+space: S000051
+property: P000010
+value: true
+---
+
+Considering the basis from the description, one may find the following:
+$$\operatorname{int}\operatorname{cl}\{2i+1\}=\operatorname{int}\{2i,2i+1,2i+2\}=\{2i+1\}$$
+$$\operatorname{int}\operatorname{cl}\{2i-1,2i,2i+1\}=\operatorname{int}\{2i-2,2i-1,2i,2i+1,2i+2\}=\{2i-1,2i,2i+1\}$$

spaces/S000051/properties/P000021.md

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+---
+space: S000051
+property: P000021
+value: false
+---
+
+The set $\{2i\mid i\in\mathbb Z\}$ is closed, discrete, and infinite.

spaces/S000051/properties/P000024.md

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+---
+space: S000051
+property: P000024
+value: true
+---
+
+Considering the basis from the description, the closure of $\{2i+1\}$ is $\{2i,2i+1,2i+2\}$,
+the closure of $\{2i-1,2i,2i+1\}$ is $\{2i-2,2i-1,2i,2i+1,2i+2\}$,
+and these sets are finite and therefore compact.

spaces/S000051/properties/P000089.md

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+---
+space: S000051
+property: P000089
+value: false
+---
+
+$x\mapsto x+2$ is a continuous map with no fixed points.

spaces/S000051/properties/P000094.md

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+---
+space: S000051
+property: P000094
+value: true
+---
+
+By construction.

spaces/S000051/properties/P000145.md

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+---
+space: S000051
+property: P000145
+value: true
+---
+
+As the {S51|P90}, the set of smallest neighborhoods (the basis given in the description)
+is a refinement of any open cover, and this collection is star-finite.

spaces/S000051/properties/P000181.md

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+---
+space: S000051
+property: P000181
+value: true
+---
+
+By construction.

spaces/S000051/properties/P000205.md

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+---
+space: S000051
+property: P000205
+value: true
+refs:
+- doi: 10.1090/s0002-9939-99-04839-x
+  name: Cut-point spaces (Honari & Bahrampour)
+---
+
+By inspection, the space is connected, and after removing a point $p$, the sets
+$\{x\mid x<p\}$ and $\{x\mid x>p\}$ are open sets partitioning $\mathbb Z\setminus\{p\}$.