3
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.