Between sequentially compactness and s-compactness
by
Ivan Gotchev
American University in Bulgaria

A T₁ space X is sequentially compact if every sequence of its points has a convergent subsequence, or - which is equivalent - every countable sequentially open cover of X has a finite subcover [3]. X is s-compact provided every sequentially open cover has a finite subcover [3]. This criterion of sequentially compactness and the different star covering properties defined and investigated in [5], [2], [1], [6] motivates the following definitions (the notation follows [6]). Let N = {1, 2, ... }. For a family of sets U in a space X and a subset A subset X we denote St(A, U) = St¹(A, U) = \cup {U in U : U \cap A =/= \emptyset}, Stⁿ⁺¹(A, U) = St(Stⁿ(A, U), U), St(x, U) = St({x}, U) and U^k = { St^k(x, U) : x in X }, k in N.

Definition

1. A space X is absolutely k-s-starcompact (k in N) if for every sequentially open cover U of X and every sequentially dense subspace Y subset X, there exists a finite subset A subset Y such that St^k(A, U) = X.
2. A space X is absolutely k-s-cl-starcompact (k in N) if for every sequentially open cover U of X and every sequentially dense subspace Y subset X, there exists a finite subset A subset Y such that [`(St^k(A, U))]^s = X.
3. A space X is absolutely weakly k-s-starcompact (k in N) if for every sequentially open cover U of X and every sequentially dense subspace Y subset X, there exists a finite subset A subset Y such that for every sequentially open neighbourhood O of A, St^k(O, U) = X.
4. A space X is absolutely weakly k-s-cl-starcompact (k in N) if for every sequentially open cover U of X and every sequentially dense subspace Y subset X, there exists a finite subset A subset Y such that for any sequentially open neighbourhood O of A, [`(St^k(O, U))]^s = X.

Theorem Every s-compact space is absolutely 1-s-starcompact.

Definition A space X is sT₂ if each pair of distinct points a, b in X belong respectively to disjoint sequentially open sets.

Theorem Absolutely 1-s-starcompact sT₂ spaces are sequentially compact.

Certain properties of the above defined absolutely s-starcompact spaces are investigated and the connection with the absolutely starcompact spaces and s-starcompact spaces [4] is discussed. Some examples and counterexamples which indicate certain of the peculiarities of absolutely s-starcompactness are presented.

References

1. E. K. van Douwen, G. M. Reed, A. W. Roscoe and I. J. Tree, Star covering properties, Topology and Appl. 39 (1991) 71-103
2. W. G. Fleischman, A new extension of countable compactness, Fund. Math. 67 (1970) 1-9.
3. I. Gotchev, On some topological properties via sequentially closed sets, talk at the International Conference on Set - Theoretical Topology and its Applications, Dec. 12 - 16 , 1994, Ehime University, Matsuyama, Japan.
4. I. Gotchev, New extensions of sequentially compactness, talk at the 13th Summer Conference on General Topology and Appl., June 24-27, 1988, National Autonomous University, Mexico City, Mexico.
5. M. V. Matveev, Absolutely countably compact spaces, Topology and Appl. 58 (1994) 81-92.
6. M. V. Matveev, A survey on star covering properties, preprint, Topology Atlas (1998), 1-136.

Date received: May 31, 1998