A subset A of X is called I-sequentially countably compact if any infinite subset A has at least one I-sequentially accumulation point in A.

A subset of X is I-sequentially compact if and only if it is I-sequentially countably compact.

Next suppose A is any I-sequentially countably compact subset of X.

A subset of X is I-sequentially compact if and only if it is sequentially countably compact in the ordinary sense.

ii) strongly countably compact if every countable cover of X by preopen sets has a finite subcover and countably S-closed [resp:countably P-closed] if every countable cover of X by regular closed [resp: preclosed] sets has a finite subcover.

iii) mildly compact (mildly countably compact, mildly Lindelof) if every clopen cover (respectively, clopen countable cover, clopen cover) of X has a finite (respectively, a finite, a countable) subcover.

mildly countably compact, mildly Lindelof), then Y is nearly compact (resp.