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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)!

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    $\begingroup$ How do you define closed sets in a topological space? $\endgroup$
    – Paul Frost
    Jan 6 '21 at 11:22
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    $\begingroup$ 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.) $\endgroup$ Jan 6 '21 at 11:24
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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.

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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

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  • $\begingroup$ 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? $\endgroup$
    – klianroeki
    Jan 6 '21 at 12:39
  • $\begingroup$ @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. $\endgroup$
    – Hermis14
    Jan 6 '21 at 13:14
  • $\begingroup$ @klianroeki I added a link for you. $\endgroup$
    – Hermis14
    Jan 6 '21 at 13:16
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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

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