Open and Closed Sets in Topological Space

Theorem
Let $T$ be a topological space.

Then $T$ and $\varnothing$ are both open and closed in $T$.

Proof
From the definition of closed set, $U$ is open in $T$ iff $T - U$ is closed in $T$.

From the definition of topology, both $T$ and $\varnothing$ are open in $T$.


 * From Set Difference Self Null, we have $T - T = \varnothing$, so $\varnothing$ is closed in $T$.


 * From Set Difference with Null, we have $T - \varnothing = T$, so $T$ is closed in $T$.