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

*

*X, empty set are closed in X

*... (not relevant for my question)

*... (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)!
 A: 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: 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
A: 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
