Description:

Definitions for form [14 AB]

Content:

Definitions for form [14 AB] (Tychonoff theorem  for sober spaces).

Definition: A topological space is sober if every closed set \(C\neq\emptyset\) that is not the union of two proper closed (not necessarily disjoint) subsets (such a \(C\) is called irreducible) is the closure of a unique singleton.

In Hausdorff spaces, irreducible closed sets are singletons and therefore Hausdorff implies sober implies \(T_0\). See Blass [1986] and Johnstone [1984].

Howard-Rubin number: 37

Type: Definitions

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