User:Cjhanrahan

Engineer with a math hobby. Welcome to my sandbox! I paste things here because the preview window doesn't render Latex in my browser so I need to publish to preview my changes :(

Ignore what you see below!

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.

Then $H^{- \circ}$ is regular open.

Proof
We must show that $H^{- \circ - \circ} = H^{- \circ}$.

First we show that $H^{- \circ - \circ} \subseteq H^{- \circ}$.

By the definition of interior, we have $H^{-0} \subseteq H^-$.

By Closure of Subset, we have $H^{- \circ -} \subseteq H^{--} = H^-$.

Applying Interior of Subset gives us $H^{- \circ - \circ } \subseteq H^-$.

Next we show $H^{- \circ - \circ} \supseteq H^{- \circ}$.

Again using the definition of interior, we have $H^{- \circ - \circ} \supseteq H^{- \circ -}$.

By Interior of Subset we have $H^{- \circ - \circ \circ} \supseteq H^{- \circ - \circ}$.