Is a family that defines a topology on any set $X$ always a set?

Is a family that defines a topology on any set $X$ always a set?

Are there families that define a topology that aren't sets?

• If you have a set $X$, then a topology is indeed a set of subsets of $X$. In fact, the power set (set of all subsets) is a set. – Thomas Sep 22 '16 at 19:01
• It is a subclass of a set, so it is a set. – Pedro Tamaroff Sep 22 '16 at 19:08
• What do you mean by "family"? Can you be specific? A topology is a set of sets. What do you mean by "family," and how does a "family" define a topology? – Thomas Andrews Sep 22 '16 at 19:09
• I'm not sure why people are voting to close as "unclear". The question is perfectly clear. Can a topology on a set ever be a proper class, and not a set? The answer is no, but the question is clear. – user223391 Sep 22 '16 at 19:22

According to your assumption, $X$ is a set. Power set of $X$ is a set. Topology is a subset of a power set; thus, it is a set.