Concerning nearly metrizable spaces

The purpose of this paper is to introduce the notion of near metrizability for topological spaces, which is strictly weaker than the concept of metrizability. A number of characterizations of nearly metrizable spaces is achieved here as analogues of the corresponding ones for metrizable spaces. It is seen that near metrizability is a natural idea vis-a-vis near paracompactness, playing the similar role as played by paracompactness with regard to metrizabilityMukherjee, MN.; Mandal, D. (2013). Concerning nearly metrizable spaces. Applied General Topology. 14(2):135-145. doi:10.4995/agt.2013.1583.SWORD135145142N. Ergun, A note on nearly paracompactness, Yokahama Math. Jour. 31 (1983), 21-25.Herrington, L. L. (1974). Properties of nearly-compact spaces. Proceedings of the American Mathematical Society, 45(3), 431-431. doi:10.1090/s0002-9939-1974-0346748-3Kovacevic, On nearly paracomapct spaces, Publ. Inst. Math. 25 (1979), 63-69.Mršević, M., Reilly, I. L., & Vamanamurthy, M. K. (1985). On semi-regularization topologies. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 38(1), 40-54. doi:10.1017/s1446788700022588M. N. Mukherjee and D. Mandal, On some new characterizations of near paracompactness and associated results, Mat. Vesnik 65 (3) (2013), 334-345.M. K. Singal and S. P. Arya, On almost regular spaces, Glasnik Mat. 4 (24) (1969), 89-99.M. K. Singal and S. P. Arya, On nearly paracompact spaces, Mat. Vesnik 6 (21) (1969), 3-16.M. K. Singal and A. Mathur, On nearly compact spaces, Boll. Un. Mat. Ital. 4 (1969), 702-710.L. A. Steen and J. A. Seebach, Counterexamples in Topology, Spinger-Verlag, New York (1970).N. V. Velicko, H-closed topological spaces, Amer. Math. Soc. Transl. 78 (1968), 103-118

Connectedness modulo a topological property

Let ${\mathscr P}$ be a topological property. We say that a space $X$ is ${\mathscr P}$-connected if there exists no pair $C$ and $D$ of disjoint cozero-sets of $X$ with non-${\mathscr P}$ closure such that the remainder $X\backslash(C\cup D)$ is contained in a cozero-set of $X$ with ${\mathscr P}$ closure. If ${\mathscr P}$ is taken to be "being empty" then ${\mathscr P}$-connectedness coincides with connectedness in its usual sense. We characterize completely regular ${\mathscr P}$-connected spaces, with ${\mathscr P}$ subject to some mild requirements. Then, we study conditions under which unions of ${\mathscr P}$-connected subspaces of a space are ${\mathscr P}$-connected. Also, we study classes of mappings which preserve ${\mathscr P}$-connectedness. We conclude with a detailed study of the special case in which ${\mathscr P}$ is pseudocompactness. In particular, when ${\mathscr P}$ is pseudocompactness, we prove that a completely regular space $X$ is ${\mathscr P}$-connected if and only if $cl_{\beta X}(\beta X\backslash\upsilon X)$ is connected, and that ${\mathscr P}$-connectedness is preserved under perfect open continuous surjections. We leave some problems open.Comment: 12 page

Monotone Versions of Countable Paracompactness

One possible natural monotone version of countable paracompactness, MCP, turns out to have some interesting properties. We investigate various other possible monotonizations of countable paracompactness and how they are related.Comment: 11 page