Boundary of Subset of Indiscrete Space

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Let $\varnothing \subsetneq H \subsetneq S$ (that is, let $H$ be a non-empty proper subset of $T$).


Then:

$\partial H = S$

where $\partial H$ denotes the boundary of $H$.


If $H = \varnothing$ or $H = S$ then $\partial H = \varnothing$.


Proof

From Closure of Subset of Indiscrete Space, $H^- = S$, where $H^-$ denotes set closure.

From Interior of Subset of Indiscrete Space, $H^\circ = \varnothing$, where $H^\circ$ denotes set interior.

By definition:

$\delta H = H^- \setminus H^\circ = S \setminus \varnothing = S$


From Open and Closed Sets in Topological Space, $\varnothing$ and $S$ are both closed and open in $T$.

So from Set Clopen iff Boundary is Empty $\delta H = \varnothing$.

$\blacksquare$


Sources