Locale of Topological Space is Locale

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Theorem

Let $T = \struct{S, \tau}$ be a topological space.

Let $\map \Omega T$ denote the locale of $T$.

Then:

$\map \Omega T$ is a locale

Proof

By definition of frame of topological space:

the frame of $T$ is $\map \Omega T$

From Frame of Topological Space is Frame:

$\map \Omega T$ is a frame

By definition of locale:

$\map \Omega T$ is a locale

$\blacksquare$

Also see

• Continuous Mapping Induces Continuous Morphism

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter II: Introduction to Locales, $\S1.1$