This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult to generalize a term if you already know how to state the original definition in terms of open sets. For example, this is one way to define the term "closed" in the context of metric spaces:(adsbygoogle = window.adsbygoogle || []).push({});

Suppose that [itex](X,d)[/itex] is a metric space. A set [itex]E\subset X[/itex] is said to beAll we have to do to generalize the term to the context of topological spaces is to change the first sentence, and leave everything else word-for-word the same:closedif [itex]E^c[/itex] is open.

Suppose that [itex](X,\tau)[/itex] is a topological space. A set [itex]E\subset X[/itex] is said to beThe only problem with this approach is that it might be possible to state the original definition in terms of open sets in more than one way. "Closed" can be equivalently defined like this:closedif [itex]E^c[/itex] is open.

Suppose that [itex](X,d)[/itex] is a metric space. A set [itex]E\subset X[/itex] is said to be(This can be thought of as a definition in terms of open sets, since we can define limits of sequences in terms of open sets). If we apply the generalization procedure discussed above to this version of the definition, what we end up with is something that certainly deserves to be called a generalization of the term "closed". But it wouldn't be equivalent with the first generalization we found. The first mathematicians who figured this out had to choose between at least three options: 1) use the first one, 2) use the second one, 3) don't generalize the term at all; instead assign different terms to the concepts associated with the two possible generalizations, for example "topologically closed" and "sequentially closed".closedif the limit of every convergent sequence in E is in E.

The consensus today is to use option 1, and to use the term "sequentially closed" for the concept associated with the second generalization. This brings me to what I want to ask. I understand why option 1 is the preferred generalization of "closed". However, it seems to me that there are always two generalizations, one that involves limits of sequences and one that doesn't, and it seems to me that we always end up choosing the one that doesn't involve sequences. So I'm wondering if there's a way to argue for this in general. Is there a way to know that the generalization that doesn't involve sequences will always be the more useful one, or did the mathematicians who chose the generalizations have to consider each term separately? Is there an example of a term for which the generalization that involves sequences has become the standard?

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# Generalizations (from metric to topological spaces)

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