Artinian spaces + basic properties

properties/P000226.md

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+---
+uid: P000226
+name: Artinian
+refs:
+  - zb: "0392.54005"
+    name: Finite $T_0$-spaces and universal mappings. (Holsztyński, Pedersen)
+---
+
+A space for which every collection of open sets has a minimal element.
+
+Equivalently:
+- Every collection of closed sets has a maximal element.
+- The open sets satisfy the *descending chain condition*: There is no infinite strictly decreasing sequence $O_1 \supsetneq O_2 \supsetneq \cdots$ of open sets.
+- The closed sets satisfy the *ascending chain condition*: There is no infinite strictly increasing sequence $Y_1 \subsetneq Y_2 \subsetneq \cdots$ of closed sets.
+
+See Section 1 of {{zb:0392.54005}}.
+
+----
+#### Meta-properties
+
+- This property is hereditary.

theorems/T000823.md

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+---
+uid: T000823
+if:
+  P000129: true
+then:
+  P000226: true
+---
+
+All collections of open sets are finite.

theorems/T000824.md

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+---
+uid: T000824
+if:
+  and:
+  - P000078: false
+  - P000002: true
+then:
+  P000226: false
+---
+
+Pick distinct $x_1,x_2,\dots\in X$. Then $\{x_1\}\subsetneq \{x_1,x_2\}\subsetneq\{x_1, x_2, x_3\} \subsetneq \dots$ is an infinite ascending chain of closed subsets.

theorems/T000825.md

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+---
+uid: T000825
+if:
+  P000078: true
+then:
+  P000226: true
+---
+
+Immediate from the definitions.

theorems/T000826.md

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+---
+uid: T000826
+if:
+  P000226: true
+then:
+  P000090: true
+---
+
+For any $x \in X$, the collection of open neighborhoods of $x$ must have a minimal element.