# How does this definition of a topology on $X$ not cause $X \in X$?

From Wikipedia:

Formally, let X be a set and let $$\tau$$ be a family of subsets of X. Then $$\tau$$ is called a topology on X if:

1. Both the empty set and X are elements of $$\tau$$.

2. Any union of elements of τ is an element of $$\tau$$.

3. Any intersection of finitely many elements of τ is an element of $$\tau$$.

I'm confused about (1). If $$\tau$$ is a family of subsets of X, and X is a member/element of $$\tau$$ then doesn't that suggest that X is a member of itself, which generally speaking isn't allowed?

• $X$ is a subset of itself, so can be a element of a family (i.e. set) of subsets of $X$ Aug 19, 2022 at 13:52

"then doesn't that suggest that X is a member of itself” No, it suggests $$X$$ is a subset of itself, because $$\tau$$ is a family of subsets of $$X$$.
Careful. A topology is a subcollection of $$2^X$$, the power set of $$X$$. So if say $$X = \{a,b\}$$, maybe $$\tau = \{\varnothing, \{a\}, \{b\}, X\}$$. Thus $$X \in \tau$$, but this says nothing about $$X \in X$$, which would, you're right, be problematic. In other words, a collection of subsets of $$X$$, such as $$\tau$$, is a different 'kind' of object than $$X$$ itself.
• I'd not seen the notation $2^X$ for the power set of $X$ before; especially if the OP is relatively new to thinking about set theory and topologies it might be a bit obstructive compared to something like $\mathcal{P}(X)$. If one is already learning to keep track of a set, its subsets and its elements, putting a notation based around functions into the mix is quite a lot. Aug 19, 2022 at 13:19
Think about the real numbers, $$\newcommand{\R}{\mathbb{R}}\R$$. Certainly this isn’t an element of itself — $$\R$$ is not a real number. The standard topology on it is given by $$\tau_\R := \{ U \subseteq \R \mid \forall x \in U,\, \exists \epsilon > 0,\ B_\epsilon(x) \subseteq U\}$$ — the collection of all open sets of reals (in the standard metric definition of “open”).
Now the biggest subset possible subset of $$\R$$ is all the reals, i.e. $$\R$$ itself, and it’s certainly open. So $$\R \in \tau_\R$$. And generally this is what the axiom says: In any topological space, the whole space (viewed as a subset of itself) is open.