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Building upon this question in Math.SE, I think the following might be rather of interest for MO.

In the literature on measure theory, probability and functional analysis the definition of a subset $A \subseteq X$ of a topological space $X$ to be relatively sequentially compact is not unique:

  1. $A$ is relatively sequentially compact in $X$ if its closure $\overline{A}$ in $X$ is sequentially compact, i.e. every sequence in $\overline{A}$ has a convergent subsequence (with limit in $\overline{A}$)
  2. $A$ is relatively sequentially compact in $X$ if every sequence in $A$ has a convergent subsequence with limit in $\overline{A}$.

In definition 2 it is not really necessary to explicitely demand the limit to be contained in $\overline{A}$ (the limit of any sequence in $A$ is always contained in $\overline{A}$ and even in the sequential closure of $A$).

Clearly, 1 $\Rightarrow$ 2 but for general topological spaces $X$ the converse does not hold as shown in the linked Math.SE thread. Another example can be also found in Meggonsin, "An Introduction to Banach Space Theory", p. 161, Exc. 2.15.

Remark: It seems to be not only the case that a few authors prefer some definition to the other but rather that roughly speaking half of the literature (I was reading so far) uses definition 1 and the other half definition 2. Both have advantages and disadvantages.

Question: Can the spaces $X$ in which 1 $\Leftrightarrow$ 2 for any subset $A \subseteq X$ be characterized by already familiar topological properties?

Note that for the weak topology on a Banach space $X$ these two definitions seem to be equivalent since I found versions of the Eberlein-Smulian theorem which equate weak relative compactness to weak relative sequential compactness with both definitions. But which property of the weak topology on $X$ is really used to equate these two definitions?

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  • $\begingroup$ The standard sources that I have seen all use definition 2. As pointed out in the linked math.so item, with definition 1 you could have a sequentially compact set that is not relatively so. And in applying the Eberlein-Smulian theorem, there may be situations where you want to assume only the property in definition 2. $\endgroup$ Jan 8, 2016 at 16:32
  • $\begingroup$ In Functional Analysis there is a widely used notion of an angelic space, introduced by Fremlin: this is a space in which a relatively countably compact space is relatively compact and compact subsets are Frechet-Urysohn. In such spaces both notions of sequential compactness seem to coincide. More information on angelic spaces can be found in the book [J.Kąkol, W.Kubiś, M.López-Pellicer, Descriptive topology in selected topics of functional analysis. Developments in Mathematics, 24. Springer, New York, 2011]. $\endgroup$ Aug 12, 2016 at 19:40

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On the matter of what definition is preferable. Let me add that the same question may be asked for the countably version, that has $\omega$-accumulation point in place of limit of a subsequence:

  1. $A$ is relatively countably compact in $X$ if its closure $\overline{A}$ in $X$ is countably compact, i.e. every sequence in $\overline{A}$ has a $\omega$-accumulation point (in $\overline{A}$).

vs

  1. $A$ is relatively sequentially compact in $X$ if every sequence in $A$ has a $\omega$-accumulation point (in $\overline{A}$).

I think 2 and 4 are more standard than the variants 1, resp. 2; in fact it seems to me there are a number of reasons to prefer them.

  • Property of language. Definitions (2) and (4) really describe relative properties, whereas (1) and (3) are just cases of the notion of sequential, resp. countable compactness, referred to the space $\overline A$.

  • Economy of language. Why squandering locutions that can be used for situations (2) and (4), while (1) and (3) can be simply referred to as “$\overline A$ is sequentially/countably compact” ?

  • Topological invariance. Properties (2) and (4) seem somehow more relevant because they behave better under continuous maps. If $A\subset X$ is (2) resp (4) in $X$, and $f:X\to Y$ is a continuous map, then $f(A)$ is (2) resp (4) in $Y$. The same is not true for the odd counterparts (1) and (3).

  • Prevalence of the situation. Here I'm vague, but I think situations (2) and (4) are quite common and relevant in the mathematical practice, because dealing with sequences taken from the set $A$ vs sequences taken from $\overline A$ can make a difference. The closure of $A$ in a weak topology may contain unknown wild objects (think of a subset $A$ of a Banach space $E$ and its closure in $E^{**}$ in the $\sigma(E^{**},E^*)$ topology, for instance).

  • Relevance in connection with important theorems. It only comes to my mind the Eberlein-Shmulian theorem, but I think this great theorem alone, a Northwest Passage of Functional Analysis, is enough to ask for a special term for situations (2) and (4).

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One important class of spaces for which the two definitions mentioned in the post are equivalent, are the first-countable spaces. One of the most important properties of any first-countable space $(X,\tau)$ is that given a subset $A\subseteq X$, a point $x$ lies in the closure of $A$ if and only if there exists a sequence $(a_n)_{n\in\mathbb{N}}$ in A which converges to $x$. Using this characterization, it is not hard to prove that definition (2) implies definition (1).

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  • $\begingroup$ Yes, this property is also satisfied for Fréchet-Urysohn spaces (a space $X$ is Fréchet-Urysohn if the sequential closure of any subset $A \subseteq X$ coincides with its closure $\overline{A}$). But does it follow from $1 \Leftrightarrow 2$ that $X$ is Fréchet-Urysohn? Recall that, an infinite-dimensional Banach space with the weak topology is not first-countable and I am not sure whether it is Fréchet-Urysohn but the fact $1 \Leftrightarrow 2$ seems to be still satisfied there. $\endgroup$
    – yada
    Jan 8, 2016 at 10:06
  • $\begingroup$ Could you add a proof that for FU spaces, (2) implies (1)? $\endgroup$ May 7, 2016 at 21:03

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