# Difference between topological space and topology

I am studying for my exam in metric and topological spaces, but got confused about two definition. I felt like they were contradicting each other. I know that a topology on X is a collection of open subsets of X which satisfy three properties. The first property is that X and the empty set are elements of the topology (from which follows that they are both open right?). But then I read further in the book and found this definition

Definition. Let X be a topological space. Then

1. X, empty set are closed in X
2. ... (not relevant for my question)
3. ... (not relevant for my question)

This really confused me, because I thought that X and the empty set had to be open sets?? Could someone please clarify this? Thank you (in advance)!

• How do you define closed sets in a topological space? Jan 6 '21 at 11:22
• A subset can be both open and closed. In particular $\emptyset$ and $X$ are both open and closed. Moreover, if you have an example of a disconnected space (e.g. $(-\infty, 0)\cup(0,+\infty)$), then there are also nontrivial examples. (In the previous example, both $(-\infty, 0)$ and $(0,+\infty)$ are both open and closed.) Jan 6 '21 at 11:24

You are wrong when you claim that “a topology on $$X$$ is a collection of open subsets of $$X$$ which satisfy three properties”. A topology on $$X$$ is a colletion $$\tau$$ of sets which satisfy three properties, and then we say that a subset $$A$$ of $$X$$ is open when (and only when) $$A\in\tau$$.

And we say that $$F\subset X$$ is closed when $$F^\complement$$ is open. So, since $$\emptyset^\complement=X$$ and $$X\in\tau$$, $$\emptyset$$ is closed. And, since $$X^\complement=\emptyset$$ and $$\emptyset\in\tau$$, $$X$$ is closed.

A subset of a topological space $$X$$ can be both open and closed. The properties are not mutually exclusive. $$\varnothing$$ and $$X$$ are open subsets of $$X$$ by definition. The closed sets are those whose complement is open. Therefore, $$\varnothing$$ and $$X$$ are also closed.

Comments) You should distinguish $$X$$ from $$\tau$$. We call a set $$X$$ together with its topology $$\tau$$ a topological space. We often write it as $$(X,\tau)$$ to clarify this. $$\tau$$ is a family of every open subset of $$X$$ whereas $$X$$ is just a set. But in many books, they just simply call $$X$$ a topological space assuming $$\tau$$ is understood in the context.

Refer to Clopen set

• Thank you! This makes it a lot clearer to me. I do have one question: can the open subsets that are in $\tau$ also be both open and closed, just like X itself? Jan 6 '21 at 12:39
• @klianroeki Of course! There can be multiple clopen sets if $X$ is disconnected. If $X$ is disconnected, by which we mean there are two non-empty disjoint open subsets of $X$ such that their union is $X$. One of such subsets is both open and closed by definition. Jan 6 '21 at 13:14
• @klianroeki I added a link for you. Jan 6 '21 at 13:16

Since $$X=\emptyset^\complement\;$$ and $$\;\emptyset=X^\complement\;$$ so in any topological space they are both closed and open sets(clopens). In a topological space these are the only clopens $$\iff$$ the space is connected