Closed Subsets and Limits of Sequences: A Topology Book Example

[Fredrik]

Anyone have a good example of a closed subset of a topological space that isn't closed under limits of sequences?

 

[micromass]

Hi Frederik!

Every closed set of a topological space is closed under limits of sequences! It's the converse that's not true. That is: there are sets which are not closed but which are still closed under limits of sequences.

For example, take the cocountable topology. Let X be a set and set

$$\mathcal{T}=\{A\subseteq X~\vert~X\setminus A~\text{is countable}\}\cup\{\emptyset\}$$

Every convergent sequence in this topology is (eventually) a constant sequence. Thus all sets are closed under limits of sequences. But not all sets are closed, of course.

Some terminology: a set that is closed under limits of sequences is called sequentially closed. A topological space where closed is equivalent with sequentially closed, is called a sequential space. As is well-known, all first-countable spaces are sequential.

 

[Fredrik]

Hi Frederik!

Hi. I actually laughed out loud when I went back here after only ten minutes and saw that you had already replied. It's appreciated, as always. (I had to go out for a while after that. I would have replied sooner otherwise).

Every closed set of a topological space is closed under limits of sequences! It's the converse that's not true.

Ah yes. I actually had that right in my mind a few minutes earlier, but somehow got it wrong anyway when I made the post. This is what I was thinking before my IQ suddenly dropped 50 points: In a metric space, a set is closed if and only if it's closed under limits of sequences. In a topological space, the corresponding statement is that a set is closed if and only if it's closed under limits of nets. Since sequences are nets, a closed set must be closed under limits of sequences. These statements suggest that there's a set E that's closed under limits of sequences and still isn't closed. Then there should exist a convergent net in E, that converges to a point in E^c. That's the sort of thing I originally meant to ask for an example of, but your example illustrates the point as well.

$$\mathcal{T}=\{A\subseteq X~\vert~X\setminus A~\text{is countable}\}\cup\{\emptyset\}$$

Every convergent sequence in this topology is (eventually) a constant sequence. Thus all sets are closed under limits of sequences.

It took me a while to understand this, but I get it now. It's a good example. It's a weird topology since even 1/n→0 is false in this topology. I think I also see an example of the kind I originally had in mind: Consider the cocountable topology on ℝ. Let E be the set of positive real numbers. Let I be the set of all open neighborhoods of 0 that have a non-empty intersection with E. Let the preorder on I be reverse inclusion. For each i in I, choose x_i in i. This defines a net in E with limit 0, which is not a member of E.

Some terminology: a set that is closed under limits of sequences is called sequentially closed. A topological space where closed is equivalent with sequentially closed, is called a sequential space. As is well-known, all first-countable spaces are sequential.

Thanks. I wasn't familiar with this terminology.

 

[micromass]

Now that I think of it, your question would actually make an ideal exam question for my topology students So that's one less question I need to come up with. Thanks a lot!

 

[wisvuze]

Hi micromass, if you remember us talking about topology books in the PF chatroom, this is discussed in the topology book by wilansky: https://www.amazon.com/dp/0486469034/?tag=pfamazon01-20

and the exact same answer/example is given too, with the cocountable topology and how every sequence would have to be eventually constant. ( it's cool! )
Not that I'm contributing much to the conversation, but I just wanted to point that out

 

[micromass]

Hi micromass, if you remember us talking about topology books in the PF chatroom, this is discussed in the topology book by wilansky: https://www.amazon.com/dp/0486469034/?tag=pfamazon01-20

and the exact same answer/example is given too, with the cocountable topology and how every sequence would have to be eventually constant. ( it's cool! )
Not that I'm contributing much to the conversation, but I just wanted to point that out

It's too bad that I can't seem to find that book anywhere I've looked around for it, because I really want to read it. (I'm actually interested in the exercises)