# Addition theorems for dense subspaces

Commentationes Mathematicae Universitatis Carolinae (2015)

- Volume: 56, Issue: 4, page 531-541
- ISSN: 0010-2628

## Access Full Article

top## Abstract

top## How to cite

topArhangel'skii, Aleksander V.. "Addition theorems for dense subspaces." Commentationes Mathematicae Universitatis Carolinae 56.4 (2015): 531-541. <http://eudml.org/doc/276210>.

@article{Arhangelskii2015,

abstract = {We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$-space. However, if a normal space $X$ is the union of a finite family $\mu $ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable by a complete metric (Theorem 2.6). We also answer a question of M.V. Matveev by proving in the last section that if a Lindelöf space $X$ is the union of a finite family $\mu $ of dense metrizable subspaces, then $X$ is separable and metrizable.},

author = {Arhangel'skii, Aleksander V.},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {dense subspace; perfect space; Moore space; Čech-complete; $p$-space; $\sigma $-disjoint base; uniform base; pseudocompact; point-countable base; pseudo-$\omega _1$-compact},

language = {eng},

number = {4},

pages = {531-541},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Addition theorems for dense subspaces},

url = {http://eudml.org/doc/276210},

volume = {56},

year = {2015},

}

TY - JOUR

AU - Arhangel'skii, Aleksander V.

TI - Addition theorems for dense subspaces

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2015

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 56

IS - 4

SP - 531

EP - 541

AB - We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$-space. However, if a normal space $X$ is the union of a finite family $\mu $ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable by a complete metric (Theorem 2.6). We also answer a question of M.V. Matveev by proving in the last section that if a Lindelöf space $X$ is the union of a finite family $\mu $ of dense metrizable subspaces, then $X$ is separable and metrizable.

LA - eng

KW - dense subspace; perfect space; Moore space; Čech-complete; $p$-space; $\sigma $-disjoint base; uniform base; pseudocompact; point-countable base; pseudo-$\omega _1$-compact

UR - http://eudml.org/doc/276210

ER -

## References

top- Arhangel'skiĭ A.V., Some metrization theorems, Uspekhi Mat. Nauk 18 (1963), no. 5, 139–145 (in Russian). MR0156318
- Arhangel'skii A.V., On a class of spaces containing all metric and all locally compact spaces, Mat. Sb. 67(109) (1965), 55–88; English translation: Amer. Math. Soc. Transl. 92 (1970), 1–39. MR0190889
- Arhangel'skii A.V., A generalization of Čech-complete spaces and Lindelöf $\Sigma $-spaces, Comment. Math. Univ. Carolin. 54 (2013), no. 2, 121–139. Zbl1289.54085MR3067699
- Arhangel'skii A.V., Choban M.M., 10.1016/j.topol.2011.03.015, Topology Appl. 159 (2012), no. 5, 1578-1590. Zbl1245.54025MR2891424DOI10.1016/j.topol.2011.03.015
- Arhangel'skii A.V., Tokgöz S., Paracompactness and remainders: around Henriksen-Isbell's Theorem, Questions Answers Gen. Topology 32 (2014), 5–15. Zbl1305.54032MR3222525
- van Douwen E.K., Tall F., Weiss W., Non-metrizable hereditarily Lindelöf spaces with point-countable bases from CH, Proc. Amer. Math. Soc. 64 (1977), 139–145. Zbl0356.54020MR0514998
- Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
- Filippov V.V., On feathered paracompacta, Dokl. Akad. Nauk SSSR 178 (1968), no. 3, 555–558. Zbl0167.21103MR0227935
- Gruenhage G., Metrizable spaces and generalizations, in: M. Hušek and J. van Mill, Eds., Recent Progress in General Topology, II, North-Holland, Amsterdam, 2002, Chapter 8, pp. 203–221. Zbl1029.54036MR1969999
- Ismail M., Szymanski A., 10.1016/0166-8641(95)90009-8, Topology Appl. 63 (1995), 69–77. Zbl0864.54001MR1328620DOI10.1016/0166-8641(95)90009-8
- Ismail M., Szymanski A., 10.1016/0166-8641(95)00082-8, Topology Appl. 71 (1996), 179–191. Zbl0864.54001MR1399555DOI10.1016/0166-8641(95)00082-8
- Ismail M., Szymanski A., 10.1016/S0166-8641(00)00043-2, Topology Appl. 114 (2001), 285–293. Zbl1012.54002MR1838327DOI10.1016/S0166-8641(00)00043-2
- Kuratowski K., Topology, vol. 1, PWN, Warszawa, 1966.
- Michael E.A., Rudin M.E., 10.2140/pjm.1977.72.497, Pacific J. Math. 72 (1977), no. 2, 497–499. MR0478093DOI10.2140/pjm.1977.72.497
- Oka S., Dimension of finite unions of metric spaces, Math. Japon. 24 (1979), 351–362. Zbl0429.54017MR0557465

## NotesEmbed ?

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