Relation between two different definitions for relative sequential compactness

Question

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?

Answer 1

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:

3. $A$ 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

4. $A$ 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).

Answer 2

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).