I am solving question $8$ of exercises in section 1.1. of chapter 1 in "Topology without tears". The question reads as follows.
Let $X$ be an infinite set and $\tau$ be a topology on $X$. If every infinite subset of $X$ is in $\tau$, prove that $\tau$ is the discrete topology
My idea was to show that every singleton set is in the topology and thereby the topology is discrete.
Choose an infinite set $A$ from $X$ such that $A^c$ is also infinite. $A$ and $A^c$ belong to $\tau$ since they are infinite sets. Now consider any $x \in X$. Note that $A \cup \{x\}$ and $A^c \cup \{x\}$ belong to $\tau$ since they are infinite sets. Hence, their intersection which is nothing but $\{x\} \in \tau$ for every $x \in X$.
The problem I have is I don't know how to prove the first sentence in the previous paragraph. These are my line of thoughts to prove them.
- If $X$ is a countably infinite set, then I can list the elements are let the odd numbered elements fall into $A$. This guarantees $A$ and $A^c$ are both infinite.
- If $X$ is uncountable, then I can choose a countably infinite subset and call it $A$.
Are the above arguments rigorous? I am not especially happy with my second argument of choosing a countably infinite subset from an uncountable set since I do not give an explicit procedure of constructing the set $A$.
Also, is there any other simpler way of answering the original question?