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?

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?

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.

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?

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?