Addition theorems, D -spaces and dually discrete spaces

Ofelia Teresa Alas; Vladimir Vladimirovich Tkachuk; Richard Gordon Wilson

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 1, page 113-124
  • ISSN: 0010-2628

Abstract

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A neighbourhood assignment in a space X is a family 𝒪 = { O x : x ∈ X } of open subsets of X such that x ∈ O x for any x ∈ X . A set Y ⊆ X is a kernel of 𝒪 if 𝒪 ( Y ) = ⋃ { O x : x ∈ Y } = X . If every neighbourhood assignment in X has a closed and discrete (respectively, discrete) kernel, then X is said to be a D -space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf P -space is a D -space and we prove an addition theorem for metalindelöf spaces which answers a question of Arhangel’skii and Buzyakova.

How to cite

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Alas, Ofelia Teresa, Tkachuk, Vladimir Vladimirovich, and Wilson, Richard Gordon. "Addition theorems, $D$-spaces and dually discrete spaces." Commentationes Mathematicae Universitatis Carolinae 50.1 (2009): 113-124. <http://eudml.org/doc/32485>.

@article{Alas2009,
abstract = {A neighbourhood assignment in a space $X$ is a family $\mathcal \{O\}= \lbrace O_x:x\in X\rbrace $ of open subsets of $X$ such that $x\in O_x$ for any $x\in X$. A set $Y\subseteq X$ is a kernel of $\mathcal \{O\}$ if $\mathcal \{O\}(Y)=\bigcup \lbrace O_x:x\in Y\rbrace =X$. If every neighbourhood assignment in $X$ has a closed and discrete (respectively, discrete) kernel, then $X$ is said to be a $D$-space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf $P$-space is a $D$-space and we prove an addition theorem for metalindelöf spaces which answers a question of Arhangel’skii and Buzyakova.},
author = {Alas, Ofelia Teresa, Tkachuk, Vladimir Vladimirovich, Wilson, Richard Gordon},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {neighbourhood assignment; $D$-space; dually discrete space; discrete kernel; scattered space; paracompactness; GO-space; neighborhood assignment; -space; dually discrete space; scattered space; -space},
language = {eng},
number = {1},
pages = {113-124},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Addition theorems, $D$-spaces and dually discrete spaces},
url = {http://eudml.org/doc/32485},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Alas, Ofelia Teresa
AU - Tkachuk, Vladimir Vladimirovich
AU - Wilson, Richard Gordon
TI - Addition theorems, $D$-spaces and dually discrete spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 1
SP - 113
EP - 124
AB - A neighbourhood assignment in a space $X$ is a family $\mathcal {O}= \lbrace O_x:x\in X\rbrace $ of open subsets of $X$ such that $x\in O_x$ for any $x\in X$. A set $Y\subseteq X$ is a kernel of $\mathcal {O}$ if $\mathcal {O}(Y)=\bigcup \lbrace O_x:x\in Y\rbrace =X$. If every neighbourhood assignment in $X$ has a closed and discrete (respectively, discrete) kernel, then $X$ is said to be a $D$-space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf $P$-space is a $D$-space and we prove an addition theorem for metalindelöf spaces which answers a question of Arhangel’skii and Buzyakova.
LA - eng
KW - neighbourhood assignment; $D$-space; dually discrete space; discrete kernel; scattered space; paracompactness; GO-space; neighborhood assignment; -space; dually discrete space; scattered space; -space
UR - http://eudml.org/doc/32485
ER -

References

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  1. Alas O.T., Junqueira L., Wilson R.G., 10.1016/j.topol.2008.04.003, Topology Appl. 155 (2008), 13 1420--1425. (2008) Zbl1169.54010MR2427413DOI10.1016/j.topol.2008.04.003
  2. Alas O.T., Sanchis M., Tkachenko M.G., Tkachuk V.V., Wilson R.G., Irresolvable and submaximal spaces: homogeneity versus σ -discreteness and new ZFC examples, Topology Appl. 107 (2000), 259--273. (2000) Zbl0984.54002MR1779814
  3. Alas O.T., Tkachuk V.V., Wilson R.G., Covering properties and neighbourhood assignments, Topology Proc. 30 1 (2006), 25--38. (2006) MR2280656
  4. Arhangel'skii A.V., 10.1090/S0002-9939-04-07336-8, Proc. Amer. Math. Soc. 132 7 (2004), 2163--2170. (2004) Zbl1045.54009MR2053991DOI10.1090/S0002-9939-04-07336-8
  5. Arhangel'skii A.V., Buzyakova R.Z., Addition theorems and D -spaces, Comment. Math. Univ. Carolin. 43 (2002), 653--663. (2002) Zbl1090.54017MR2045787
  6. Arhangel'skii A.V., Collins P.J., 10.1016/0166-8641(94)00093-I, Topology Appl. 64 3 (1995), 219--241. (1995) MR1342519DOI10.1016/0166-8641(94)00093-I
  7. Buzyakova R.Z., Tkachuk V.V., Wilson R.G., A quest for nice kernels of neighbourhood assignments, Comment. Math. Univ. Carolin. 48 4 (2007), 689--697. (2007) Zbl1199.54141MR2375169
  8. van Douwen E.K., Lutzer D., 10.1090/S0002-9939-97-03902-6, Proc. Amer. Math. Soc. 125 4 (1997), 1237--1245. (1997) Zbl0885.54023MR1396999DOI10.1090/S0002-9939-97-03902-6
  9. van Douwen E., Pfeffer W.F., 10.2140/pjm.1979.81.371, Pacific J. Math. 81 2 (1979), 371--377. (1979) MR0547605DOI10.2140/pjm.1979.81.371
  10. van Douwen E., Wicke H.H., A real, weird topology on the reals, Houston J. Math. 3 1 (1977), 141--152. (1977) Zbl0345.54036MR0433414
  11. Eisworth T., On D -spaces, in Open Problems in Topology, II, ed. Elliott Pearl, Elsevier, Amsterdam, 2007. MR2367385
  12. Engelking R., General Topology, Heldermann, Berlin, 1989. Zbl0684.54001MR1039321
  13. Galvin F., Indeterminacy of point-open games, Bull. Acad. Polon. Sci., Sér. Math. 26 5 (1978), 445--449. (1978) Zbl0392.90101MR0493925
  14. Gruenhage G., A note on D -spaces, Topology Appl. 153 (2006), 2229--2240. (2006) Zbl1101.54029MR2238727
  15. Lutzer D., Ordinals and paracompactness in ordered spaces, Proceedings Topo72, General Topology and its Applications, Pittsburgh International Conference, 1972, Lecture Notes in Mathematics 372, Springer, Berlin, 1974. Zbl0298.54015MR0362250
  16. van Mill J., Tkachuk V.V., Wilson R.G., 10.1016/j.topol.2006.03.029, Topology Appl. 154 (2007), 2127--2134. (2007) Zbl1131.54022MR2324924DOI10.1016/j.topol.2006.03.029
  17. Telgársky R., Spaces defined by topological games, Fund. Math. 88 (1975), 193--223. (1975) MR0380708
  18. Telgársky R., Spaces defined by topological games II, Fund. Math. 116 (1983), 189--207. (1983) MR0716219
  19. Uspenskij V.V., The frequency spectrum of function spaces (in Russian), Vestnik. Moskov. Univ. Ser. I Mat. 37 1 (1982), 31--35. (1982) 

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