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).

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).

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) behave better. 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).

• Usefulness of the notion. In the mathematical practice, dealing with sequences from the set $A$ vs sequences in $\overline A$ 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).

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).