Self-studying Folland Chapter 4 Point Set Topology

A topology on $X$ is a family $\mathcal{T}$ of subsets of $X$ that contains $\emptyset$ and $X$ and is closed under arbitrary unions and finite intersections.

Members of $\mathcal{T}$ are called open sets. Their complements are called closed sets. I don't think it's necessary that closed sets are in the topology? Or can there actually be closed sets in topology?

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    $\begingroup$ There can be, but usually not all closed sets are in the topology, and very often none other than the empty set and the whole space are. $\endgroup$ – Matt Samuel Dec 20 '16 at 11:36
  • $\begingroup$ If $\emptyset$ belong to topology, it's open by definition. Its complement is $X$, so $X$ is closed. However, $X$ belongs to the topology, too. So closed sets may belong to topology. $\endgroup$ – CiaPan Dec 20 '16 at 11:37
  • $\begingroup$ Depending on the space, a closed set can be open. In fact, $\emptyset$ and $X$ are. Some people call "clopens" to these sets. $\endgroup$ – ajotatxe Dec 20 '16 at 11:41
  • $\begingroup$ @ZHU, you have asked 45 questions but accepted answers to only 6 of them. While nobody is expected to accept answers to all the asked questions, 6 out of 45 is not reasonable. Please review your questions and look for acceptable answers, I am sure you will find some. $\endgroup$ – Alex M. Dec 20 '16 at 15:45

Indeed, closed sets usually do not belong to the topology. Nevertheless, there are cases when they do. One such example is of $\mathcal T = \mathcal P (X)$, the set of all the subsets of $X$. Another example is $\mathcal T = \{ \emptyset, X \}$.

Furthermore, $\emptyset$ and $X$ are always both closed and open, so they are examples of closed sets belonging to $\mathcal T$.

Every connected component of $X$ is both closed and open.

  • $\begingroup$ Since they are in the topology by definition, a topology always contains a closed set? $\endgroup$ – ZHU Dec 20 '16 at 11:41
  • $\begingroup$ Yes, $\emptyset$ and $X$ are by definition always in the topology. And you can prove that they are both open and closed. $\endgroup$ – Jakob Elias Dec 20 '16 at 11:44

Depending on the space, a closed set can be open. In fact, $\emptyset$ and $X$ always are, but there are more examples. Some people call these sets "clopen sets". A clopen set and its complement describe in some sense a splitting of the space. Spaces that have non trivial clopen sets are called "not connected".


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