Addition theorems and $D$-spaces

Arhangel'skii, Aleksander V., and Buzyakova, Raushan Z.. "Addition theorems and $D$-spaces." Commentationes Mathematicae Universitatis Carolinae 43.4 (2002): 653-663. <http://eudml.org/doc/249004>.

@article{Arhangelskii2002,
abstract = {It is proved that if a regular space $X$ is the union of a finite family of metrizable subspaces then $X$ is a $D$-space in the sense of E. van Douwen. It follows that if a regular space $X$ of countable extent is the union of a finite collection of metrizable subspaces then $X$ is Lindelöf. The proofs are based on a principal result of this paper: every space with a point-countable base is a $D$-space. Some other new results on the properties of spaces which are unions of a finite collection of nice subspaces are obtained.},
author = {Arhangel'skii, Aleksander V., Buzyakova, Raushan Z.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$D$-space; point-countable base; extent; metrizable space; Lindelöf space; -space; point-countable base; extent; metrizable space; Lindelöf space},
language = {eng},
number = {4},
pages = {653-663},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Addition theorems and $D$-spaces},
url = {http://eudml.org/doc/249004},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
AU - Buzyakova, Raushan Z.
TI - Addition theorems and $D$-spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 4
SP - 653
EP - 663
AB - It is proved that if a regular space $X$ is the union of a finite family of metrizable subspaces then $X$ is a $D$-space in the sense of E. van Douwen. It follows that if a regular space $X$ of countable extent is the union of a finite collection of metrizable subspaces then $X$ is Lindelöf. The proofs are based on a principal result of this paper: every space with a point-countable base is a $D$-space. Some other new results on the properties of spaces which are unions of a finite collection of nice subspaces are obtained.
LA - eng
KW - $D$-space; point-countable base; extent; metrizable space; Lindelöf space; -space; point-countable base; extent; metrizable space; Lindelöf space
UR - http://eudml.org/doc/249004
ER -

References

top

Arens R., Dugundji J., Remark on the concept of compactness, Portugal. Math. 9 (1950), 141-143. (1950) Zbl0039.18602 MR0038642
Arhangel'skii A.V., Buzyakova R.Z., On some properties of linearly Lindelöf spaces, Topology Proc. 23 (1998), 1-11. (1998) Zbl0964.54018 MR1800756
Balogh Z., Gruenhage G., Tkachuk V., Additivity of metrizability and related properties, Topology Appl. 84 (1998), 91-103. (1998) Zbl0991.54032 MR1611277
Boone J.R., On irreducible spaces, 2, Pacific J. Math. 62.2 (1976), 351-357. (1976) MR0418037
Borges C.R., Wehrly A.C., A study of $D$ -spaces, Topology Proc. 16 (1991), 7-15. (1991) Zbl0787.54023 MR1206448
Burke D.K., Covering properties, in: K. Kunen and J. Vaughan, Eds, Handbook of Set-theoretic Topology, Chapter 9, pp.347-422; North-Holland, Amsterdam, New York, Oxford, 1984. Zbl0569.54022 MR0776628
Buzyakova R.Z., On $D$ -property of strong $Σ$ -spaces, Comment. Math. Univ. Carolinae 43.3 (2002), 493-495. (2002) Zbl1090.54018 MR1920524
de Caux P., A collectionwise normal, weakly $θ$ -refinable Dowker space which is neither irreducible nor realcompact, Topology Proc. 1 (1976), 66-77. (1976) Zbl0397.54019
Christian U., Concerning certain minimal cover refinable spaces, Fund. Math. 76 (1972), 213-222. (1972) MR0372818
van Douwen E., Pfeffer W.F., Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81.2 (1979), 371-377. (1979) Zbl0409.54011 MR0547605
van Douwen E.K., Wicke H.H., A real, weird topology on reals, Houston J. Math. 13.1 (1977), 141-152. (1977) MR0433414
Ismail M., Szymanski A., On the metrizability number and related invariants of spaces, 2, Topology Appl. 71.2 (1996), 179-191. (1996) MR1399555
Ismail M., Szymanski A., On locally compact Hausdorff spaces with finite metrizability number, Topology Appl. 114.3 (2001), 285-293. (2001) Zbl1012.54002 MR1838327
Michael E., Rudin M.E., Another note on Eberlein compacts, Pacific J. Math. 72 (1977), 497-499. (1977) Zbl0344.54018 MR0478093
Ostaszewski A.J., Compact $σ$ -metric spaces are sequential, Proc. Amer. Math. Soc. 68 (1978), 339-343. (1978) MR0467677
Rudin M.E., Dowker spaces, in: K. Kunen and J. Vaughan, Eds, Handbook of Set-theoretic Topology, Chapter 17, pp.761-780; North-Holland, Amsterdam, New York, Oxford, 1984. Zbl0566.54009 MR0776636
Tkachenko M.G., On compactness of countably compact spaces having additional structure, Trans. Moscow Math. Soc. 2 (1984), 149-167. (1984)
Wicke H.H., Worrell J.M., Jr., Point-countability and compactness, Proc. Amer. Math. Soc. 55 (1976), 427-431. (1976) Zbl0323.54013 MR0400166
Worrell J.M., Wicke H.H., Characterizations of developable spaces, Canad. J. Math. 17 (1965), 820-830. (1965) MR0182945
Worrell J.M., Jr., Wicke H.H., A covering property which implies isocompactness. 1, Proc. Amer. Math. Soc. 79.2 (1980), 331-334. (1980) MR0565365

Citations in EuDML Documents

top

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Language to use for this widget.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Number of notes per page

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.