Countably many isolated points implies that any disjoint union of open sets is countable.

Question

I have a problem in general topology that I am not able to solve.

Let $X$ be a topological space with denumerably infinite many isolated points. Then show that any collection of pairwise disjoint open sets is denumerably infinite.

Any help would be much appreciated.

Answer 1

Unless I'm missing something, the claim is false.

Let $B$ be any uncountable collection of pairwise disjoint sets, each containing at least two points, and let $X$ be the union of the members of $B$.

Let $T$ be the set of all unions of any number of members of $B.\;\,$Then

• $T$ is a topology on $X$.
• There are no isolated points.
• There is an uncountable collection of pairwise disjoint open sets, namely, the members of $B$.

For a concrete example, let $X = \mathbb{R}-\{0\}$, and let $B = \{\{x,-x\} \mid x > 0\}$. Thus, the topology $T$ will be comprised of all subsets $U$ of $X$ such that whenever $x$ in $U,\;-x$ is also in $U$.

Edit:

The original version of the question specified countably many isolated points, so my counterexample qualifies, since it has no isolated points.

But in the OP's new edit, the requirement is now a countably infinite number of isolated points, not just countable.

But it's easy to satisfy this new requirement . . .

Just take any countably infinite set $S$, where $S$ is disjoint from $X$, and let $Y = X \cup S$.

Then define a topology on $Y$ by declaring a subset of $Y$ as open if it's the union of a (possibly empty) subset of $S$ with an open (possibly empty) subset of $X$, where "open" means open in the topology previously defined on $X$.

Then $Y$ is a topological space with a countably infinite number of isolated points, namely the points of $S$, and there is an uncountable collection of pairwise disjoint open subsets of $Y$, namely, the members of $B$.

Alternatively, here's a concrete example, similar to the old one, which satisfies the new requirements . . .

Let $X=\mathbb{R}$, and declare a set open if it's the union of a (possibly empty) subset of $\mathbb{Z}$ with a (possibly empty) subset $U$ of $\mathbb{R}-\mathbb{Z}$ such that whenever $x$ is in $U,\;-x$ is also in $U$.

Then the set of isolated points of $X$ is the set $\mathbb{Z}$, which is countably infinite, and the set $$B = \{\{x,-x\} \mid x\in \mathbb{R}-\mathbb{Z}\;\text{and}\;x > 0\}$$ is an uncountable collection of pairwise disjoint open subsets of $X$.