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Anyone have a good example of a closed subset of a topological space that isn't closed under limits of sequences?
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).micromass said:Hi Frederik!
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 Ec. That's the sort of thing I originally meant to ask for an example of, but your example illustrates the point as well.micromass said:Every closed set of a topological space is closed under limits of sequences! It's the converse that's not true.
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 xi in i. This defines a net in E with limit 0, which is not a member of E.micromass said:[tex]\mathcal{T}=\{A\subseteq X~\vert~X\setminus A~\text{is countable}\}\cup\{\emptyset\}[/tex]
Every convergent sequence in this topology is (eventually) a constant sequence. Thus all sets are closed under limits of sequences.
Thanks. I wasn't familiar with this terminology.micromass said: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.
wisvuze said: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
A sequence in mathematics is a list of numbers or objects that follow a specific pattern or rule. Each individual element in the sequence is called a term, and the position of the term in the sequence is known as its index. Sequences can be finite, meaning they have a specific number of terms, or infinite, meaning they continue indefinitely.
Sequences are used in real life in a variety of ways. One common example is in the stock market, where stock prices are often represented as a sequence of numbers over time. Other examples include weather patterns, population growth, and computer algorithms.
A closed set in mathematics is a set that contains all of its limit points. In other words, if a sequence of points in a closed set approaches a point outside of the set, that point must also be included in the set.
In topology, closed sets are used to define the concept of continuity. A function between topological spaces is considered continuous if the inverse image of every closed set in the range is a closed set in the domain. Closed sets are also used to define other important concepts in topology, such as compactness and connectedness.
Sequences and closed sets are closely related in mathematics. In particular, a sequence can be used to define a closed set. For example, the set of limit points of a sequence is always a closed set. On the other hand, a closed set can also be used to define a sequence, as the terms in the sequence can be seen as approaching the limit points of the closed set.