In the definition of a topology, must the index set for unions of open sets be countable?

Question

I have the following definition.

Let $X$ be a set. A topology on $X$ is a family $\tau$ of subsets of $X$ such that

(i) $\emptyset,X\in\tau$

(ii) if $U_i\in \tau$ for all $i\in I$ where $I$ is some index set, then $\bigcup_{i\in I}U_i\in\tau$; and

(iii) if $U,V\in\tau$ then $U\cap V\in\tau$.

(Quick) Question: In (ii), need the index set be countable?

No, in fact in a lot of cases it's very important to allow it to be uncountable — Alex Pawelko, Nov 12 '22 at 21:25
No, any union of open sets is open. You could rewrite (ii) as any $\mathcal U\subseteq \tau$ has $\bigcup_{U\in\mathcal U} U\in \tau.$ — Thomas Andrews, Nov 12 '22 at 21:25
In general no, consider topology on $\mathbb{R}$ induced by its metric — Sine of the Time, Nov 12 '22 at 21:25
Thank you all. If one of you can post an answer with your comment (with a citation possibly) I will emmidietly accept it. — Maths Wizzard, Nov 12 '22 at 21:26

score 2 · Accepted Answer · answered Nov 12 '22 at 21:43

(The comments answered this, just to take this out of the unanswered queue)

No, there are no restrictions on what the index set must be in the union in the definition of a topology - this is commonly stated as the fact that the arbitrary union of open sets is open.

In fact, there arise many situations where it is important that one is allowed to take an uncountable union of open sets, see some discussion about the immediate difference it makes here Is there a difference between allowing only countable unions/intersections, and allowing arbitrary (possibly uncountable) unions/intersections? and What is the motivation behind the arbitrary union topological axiom? (There are many many more instances though where this comes up once you do more topology, for instance the distinction between finite/countable/uncountable union becomes a very important distinction when talking about things like Lindelof spaces and compactness)

In the definition of a topology, must the index set for unions of open sets be countable?

1 Answers1