Prove that is a topology

Question

Let X be any set having more that one element . Pick $ a \in X $ , then fix it. The one-point topology on X is $τ_a $ = {∅,X, {a}} Check that $ τ_a $ is indeed a topology on X

Answer 1

You have to prove that it satisfies the axioms of a topology, that are :

1) $\emptyset$, $X$ are open sets.

2) Any union of open sets is an open set.

3) Any finite intersection of open sets is an open set.

Here, you indeed have $\emptyset, X \in \tau_a$, so $1)$ is ok.

Then, all the possible unions $\emptyset \cup \lbrace a \rbrace = \lbrace a \rbrace$, $\emptyset \cup X = X$ and $X \cup \lbrace a \rbrace = X$ are in $\tau_a$ so $2)$ is ok.

And finally, all the possible intersections $\emptyset \cap \lbrace a \rbrace = \emptyset$, $\emptyset \cap X = \emptyset$ and $X \cap \lbrace a \rbrace = \lbrace a \rbrace$ are in $\tau_a$ so $3)$ is ok.