Strongly KC-Spaces, Expandability And D_σ-Spaces

The notion of KC-spaces was first introduced by A. Wilansky in 1967. A topologicalspace X is called a KC-space if every compact subset of X is closed. KC-spaces can be regarded as a seperate property between T₁ and T₂. One of the old questions on KC-spaces posed by R. Larson is whether a space (X,Ï„) is maximal compact if and only if it is minimal KC. Many authors have investigated this problem. However, up to now, Larson's original question remains open. A related question to Larson's is whether every KC-space is Katetov-KC, that is, whether every KC topology contains a minimal KC topology. W. Fleissner constructed an example of a KC-space which is not Katetov-KC. 0. T. Alas and R. G. Wilson gave a characterization of Katetov-KC spaces and at the end of the paper, they asked whether every countably compact KC-spa.ce of size less than c has the FDS-property.Expandability is an important concept in general topology. L. L. Krajewski obtainedvarious results relating this property with certain topological covering properties and proved that every paracompact space is expandable. Smith and Krajewski introducedvarious generalizations of the expandability and showed that a metacompact space is almost expandable. Hodel proved that every T\ wN-space is also almost expandable.C. Good, R. Kinight and I. Stares further asked whether a wN-space is expandable. Chris Good, Daniel Jennings and Abdul M. Mohamad introduced the notion of symmetric g-function and showed that every sym-wg space is wN. They also showed that a normal sym-wg T₁ space is expandable. Recently K. Y. Al-zoubi introduceda new kind of expandability-s-expandability and gave a characterization ofω₀-s-expandability for extremally disconnected spaces.The class of D-spaces was intoduced by van Douwen in 1979. A principal property of D-spaces is that the extent coincides with the Lindelof number in such spaces. In particular, every countably compact D-space is compact and every D-space with the countable extent is Lindelof. These facts make the notion of a D-space useful in studying covering properties. It is not easy to verify whether a space is a D-space. A. V. Arhangel'skii considered a formally weaker property which is much easier to verify for large classes of spaces and is still strong enough to imply compactness for countably compact spaces. Peng Liangxue introduced another generalization of the class of D-spaces-D_σ-spaces. It is easy to see that every Lindelof space is a D_σ-space.As the further study of the above subjects, we mainly investigate problems of three aspects. The main results are as follows:In the first chapter, a new kind of KC-spaces is investigated, that is to say the class of spaces in which all countably compact subsets are closed. Such spaces are said to be strongly KC-spaces and they have good properties. In section 1.3, the relationship between strongly KC and KC-spaces is discussed. In section 1.4, the products of strongly KC-spaces and the one point countable compactification of a strongly KC-space are studied. Then in section 1.5, we mainly prove that, for strongly KC-spaces and maximal countably compact spaces, there is a positive answer to a question analogous to Larson's, that is, a space (X,Ï„) is maximal countably compact if and only if it is minimal strongly KC. Note that the equivalence here require no restriction on spaces. Applying this we prove that minimal strongly KC-spaces are closed hereditarily and study other properties of minimal strongly KC-spaces, Finally some sufficient conditions for a space to have the FDS-property are further studied, specially, as a corollary, we prove that a Lindelof T₂ space with countable pseudocharacter has the FDS-property, which gives a partial answer to a question of O. T. Alas, M. G. Tkachenko and V. V. Tkachenuk. In section 1.6, we first prove that every countably compact KC-space of cardinality lees than c has the FDS-property, which answers positively a question of O. T. Alas and R. G. Wilson. Then using this result we give a characterization for a space (X,Ï„) to be K atetov-strongly KC and present a sufficient condition of Katetov-strongly KC-spaces.In Chapter two, we mainly deal with the expandability of wN-spaces and the characterization of s-expandability. First in section 2.3, it is proved that every T₁ wN-space is expandable and thus a question given by Chris Good in 2000 is answered positively and consequently some known results are extended. Then in section 2.4, we investigate s-expandability and for extremally disconnected spaces, a characterization of s-expandability is given in terms of covers, which extends one known result and finally a sufficient condition for a T₂ space to be s-paracompact is provided.In Chapter three, a generalization of D-spaces-D_σ-spaces is studied. In section 3.3, we study some properties of D_σ-spaces, extend some results concerning D-spaces to D_σ-spaces and prove that, forθ- refinable spaces, local D_σimplies global D_σ.