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 $$T$$.