Provide an example of a first countable space $X$ and an uncountable subset $A$ of $X$ with no accumulation point

Question

I've been thinking about the following:

Provide an example of a first countable space $X$ and an uncountable subset $A$ of $X$ with no accumulation point.

I've never been good at coming up with examples of topological spaces exhibiting particular properties. The first part of this problem was to show that if $X$ is in addition second countable then $A$ will always have an accumulation point.

I thought about a couple of the classic counterexample spaces, but they don't seem to work.

I wish I could give more information as to what I've tried etc., but most of my work so far has just been trying out different spaces, which is hard to write down. Could anyone point in the right direction? Any help is appreciated. Thanks

Answer 1

Let $X$ be an uncountable set and consider the discrete topology on it. The $X$ is an uncountable subset of $X$ without accumulation points.

Answer 2

Let X be an uncountable discrete space and A any uncountable shbset of X.