when compactness implies finiteness

Question

Let $X$ be a topological space. It is known that if $A\subset X$ is finite then $A$ is compact. I want to know what condition must be added such that the converse holds, that is if $A$ is compact then $A$ is finite. I think that this condition holds whenever $X$ is discrete space.

What are other conditions such that every compact set is finite?

Is it true that if every compact space $X$ is finite then $X$ must be discrete space?

Answer 1

The co-countable topology on an uncountable set $X$ also has this property. (The nonempty open sets are those $U \subseteq X$ such that $X \setminus U$ is countable.)

• Proof. If $A \subseteq X$ is infinite, take a countably infinite subset $\{ x_i : i \in \mathbb N \}$ of $A$. Then for each $j$ the set $U_j = X \setminus \{ x_i : i \neq j \}$ is an open set, and $\bigcup_j U_j \supseteq A$. Note that $x_i \in U_j$ iff $i = j$. From this it follows that given $j_1, , \ldots , j_n$ we have $U_{j_1} \cup \cdots \cup U_{j_n} \not\supseteq A$.