1

I know that in metric spaces sequences capture the properties of the space, and in general topological nets capture the properties of the space. My question is when can we definitely know that sequences capture the properties of the topological space?

I thought that being a first coutable space is enough, but have recently been told that this is not the case with compactness and sequential compactness for example. Can anyone clarify whether there is a general condition for a space being described by sequences?

Keen-ameteur
  • 8,404
  • 1
    when can we definitely know that sequences capture the properties of the topological space? --- See the Wikipedia article sequential space. Also, a wider class of spaces, but not ALL spaces, can be characterized if we replace ordinary sequences with arbitrary transfinite ordinal sequences (in trying to overcome lack of first countability) --- is a net stronger than a transfinite sequence for characterizing topology? AND Intuition behind nets. – Dave L. Renfro Feb 12 '20 at 07:56
  • So just to verify, according to the wikipedia article a first countable space is in fact a sequential space, and hence is characterizable by sequences? – Keen-ameteur Feb 12 '20 at 09:55
  • 1
    Yes, every first countable space is a sequential space. However, there are sequential spaces that are not first countable spaces. For the analogs in the case of transfinite ordinal sequences (i.e. when you use ordinals to obtain longer "sequences" than possible using ordinary indexed-by-natural-numbers sequences), the relevant terms are local character (or character at a point, (continued) – Dave L. Renfro Feb 12 '20 at 10:27
  • 1
    which generalizes first countability to higher cardinalities, and radial spaces, which generalizes sequential spaces to various ways of dealing with sequences of transfinite lengths. – Dave L. Renfro Feb 12 '20 at 10:28
  • Could you write your comments as an answer, so I can tag this thread as answered? – Keen-ameteur Feb 12 '20 at 11:29
  • OK, the comments (collectively) are probably appropriate now. I had been thinking Martin Sleziak would jump in and say way more than anything I could on this topic, but perhaps he's been busy with something else. – Dave L. Renfro Feb 12 '20 at 11:53
  • 1
    Note that even in a first countable space sequential compactness can never give more than countable compactness, generally not full compactness (in metric spaces you can but that's a special case). Lindelöfness is not characterised by sequence-like properties... – Henno Brandsma Feb 12 '20 at 22:48

1 Answers1

2

when can we definitely know that sequences capture the properties of the topological space? --- See the Wikipedia article sequential space. Also, a wider class of topological spaces, but not ALL topological spaces, can be characterized if we replace ordinary sequences with arbitrary transfinite ordinal sequences --- see is a net stronger than a transfinite sequence for characterizing topology? AND Intuition behind nets. The use of transfinite ordinal sequences is a natural way to overcome the lack of first countability, but it turns out that more than just higher-cardinality "countability conditions" are needed to capture the notion of a point being in the closure of a set. That is, for topological spaces in general, being able to capture the notion of a point being in the closure of a set takes more than just having "long enough sequences to reach the point".

To summarize, every first countable space is a sequential space. However, there are sequential spaces that are not first countable spaces. For the analogs in the case of transfinite ordinal sequences (i.e. when you use ordinals to obtain longer "sequences" than is possible using ordinary indexed-by-natural-numbers sequences), the relevant terms are local character (or character at a point), which generalizes first countability to higher cardinalities, and radial space, which generalizes sequential spaces to various ways one can deal with sequences of transfinite lengths.

Dave L. Renfro
  • 38,688