Topology with Set Inclusion Forms a Frame/Corollary

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Theorem

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

Let $\map \Omega T = \struct{\tau, \subseteq}$ be the topology $\tau$ with set inclusion.

Then:

$\map \Omega T$ is a locale

Proof

Follows immediately from:

• Definition:Locale (Lattice Theory)
• Topology with Set Inclusion Forms a Frame

$\blacksquare$

Also see

• Continuous Mapping Induces Continuous Morphism

Sources

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