On spaces which are linearly \(D\). (English) Zbl 1180.54009

Guo, Hongfeng; Junnila, Heikki

Topology Appl. 157, No. 1, 102-107 (2010).

Summary: We introduce a generalization of \(D\)-spaces, which we call linearly \(D\)-spaces. The following results are obtained for a \(T_{1}\)-space \(X\).

–

\(X\) is linearly Lindelöf if, and only if, \(X\) is a linearly \(D\)-space of countable extent.

–

\(X\) is linearly \(D\) provided that \(X\) is submetaLindelöf.

–

\(X\) is linearly \(D\) provided that \(X\) is the union of finitely many linearly \(D\)-subspaces.

–

\(X\) is compact provided that \(X\) is countably compact and \(X\) is the union of countably many linearly \(D\)-subspaces.

MSC:

54A25	Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)

Keywords:

\(D\)-space; submetaLindelöf; linearly Lindelöf; countably compact

References:

[1]	Arhangel’skii, A.V., Topological function spaces, (1992), Kluwer Academic Publishers Dordrecht · Zbl 0911.54004
[2]	Arhangel’skii, A.V., D-spaces and finite unions, Proc. amer. math. soc., 132, 2163-2170, (2004) · Zbl 1045.54009
[3]	Arhangel’skii, A.V., D-spaces and covering properties, Topology appl., 146/147, 437-449, (2005) · Zbl 1063.54013
[4]	Arhangel’skii, A.V.; Buzyakova, R.Z., On linearly Lindelöf and strongly discretely Lindelöf spaces, Proc. amer. math. soc., 127, 2449-2458, (1999) · Zbl 0930.54003
[5]	Arhangel’skii, A.V.; Buzyakova, R.Z., Addition theorems and D-spaces, Comment. math. univ. carolin., 43, 653-663, (2002) · Zbl 1090.54017
[6]	Aull, C.E., A generalization of a theorem of aquaro, Bull. austral. math. soc., 9, 105-108, (1973) · Zbl 0255.54015
[7]	Balogh, Z.; Rudin, M.E., Monotone normality, Topology appl., 47, 115-127, (1992) · Zbl 0769.54022
[8]	Borges, C.R.; Wehrly, A.C., A study of D-spaces, Topology proc., 16, 7-15, (1991) · Zbl 0787.54023
[9]	Buzyakova, R.Z., Hereditary D-property of function spaces over compacta, Proc. amer. math. soc., 132, 2171-2181, (2004) · Zbl 1053.54038
[10]	Buzyakova, R.Z.; Tkachuk, V.V.; Wilson, V.V., A quest for Nice kernels of neighbourhood assignments, Comment. math. univ. carolin., 48, 689-697, (2007) · Zbl 1199.54141
[11]	van Douwen, E.K., Why certain čech – stone remainders are not homogeneous, Collect. math., 41, 45-52, (1979) · Zbl 0424.54012
[12]	van Douwen, E.K.; Lutzer, D.J., A note on paracompactness in generalized ordered spaces, Proc. amer. math. soc., 125, 1237-1245, (1997) · Zbl 0885.54023
[13]	van Douwen, E.K.; Pfeffer, W., Some properties of the sorgenfrey line and related spaces, Pacific J. math., 81, 371-377, (1979) · Zbl 0409.54011
[14]	van Douwen, E.K.; Wicke, H.H., A real, weird topology on the reals, Houston J. math., 13, 141-152, (1977) · Zbl 0345.54036
[15]	Dow, A.; Junnila, H.J.K.; Pelant, J., Coverings, networks and weak topologies, Mathematika, 53, 287-320, (2006) · Zbl 1138.46012
[16]	Gerlitz, J.; Juhasz, I.; Szentmiklossy, Z., Two improvements of Tkacenko’s addition theorem, Comment. math. univ. carolin., 46, 705-710, (2005) · Zbl 1121.54041
[17]	Gruenhage, G., A note on D-spaces, Topology appl., 153, 2229-2240, (2006) · Zbl 1101.54029
[18]	Junnila, H.J.K., Neighbournets, Pacific J. math., 76, 83-108, (1978)
[19]	Kunen, K., Locally compact linearly Lindelöf spaces, Comment. math. univ. carolin., 43, 155-158, (2002) · Zbl 1090.54019
[20]	Peng, L.-X., A note on D-spaces and infinite unions, Topology appl., 154, 2223-2227, (2007) · Zbl 1133.54012
[21]	Mischenko, A.S., Finally compact spaces, Soviet math. dokl., 145, 1199-1202, (1962) · Zbl 0121.17501