×

Relative normality and dense subspaces. (English) Zbl 1008.54013

A subspace \(Y\) of a topological space \(X\) is said to be normal in \(X\) if for every pair \(A,B\) of disjoint closed subsets of \(X\) there exist disjoint open subsets \(U,V\) of \(X\) such that \(A\cap Y\subset U\) and \(B\cap Y\subset V\). It is said to be internally normal in \(X\) if for every pair \(A,B\) of disjoint closed subsets of \(X\) such that \(A\subset Y\) and \(B\subset Y\) there exist disjoint open subsets \(U,V\) of \(X\) such that \(A\subset U\) and \(B\subset V\). If \(Y\) is normal in \(X\), then \(Y\) is internally normal in \(X\). Moreover, every dense normal subspace of a topological space \(X\) is internally normal in \(X\). The paper contains some interesting examples concerning these relative normality properties.
Example 1: There exists a completely regular \(T_1\)-space \(X\) containing a dense subspace \(Y\) such that \(Y\) is internally normal in \(X\) but not normal in \(X\).
Example 2: The space \(C_p(\omega_1+1)\) of all continuous real-valued functions on \(\omega_1+1\) equipped with the topology of pointwise convergence has the property that for every dense subspace \(X\) of \(C_p (\omega_1+1)\) and for every dense subspace \(Y\) of \(X\), \(Y\) is not internally normal in \(X\). It follows that \(C_p(\omega_1+1)\) has no dense normal subspace.
Example 3: There exists a submetrizable completely regular \(T_1\)-space \(X\) of weight \(\omega_1\) such that no dense subspace of \(X\) is internally normal in \(X\). It follows that \(X\) has no dense normal subspace.
Example 4: The space \({\mathcal F}(\mathbb{R})\) of all nonempty finite subsets of the reals equipped with the Pixley-Roy topology is a completely regular hereditarily metacompact and developable \(T_1\)-space such that no dense subspace of \({\mathcal F}(\mathbb{R})\) is internally normal in \({\mathcal F}(\mathbb{R})\). Hence \({\mathcal F}(\mathbb{R})\) has no dense normal subspace.

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54C35 Function spaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arhangel’skii, A. V., Structure and classification of topological spaces and cardinal invariants, Russian Math. Surveys, 33, 33-96 (1978) · Zbl 0428.54002
[2] Arhangel’skii, A. V., Topological Function Spaces (1992), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0911.54004
[3] Arhangel’skii, A. V., Relative topological properties and relative topological spaces, Topology Appl., 70, 87-99 (1996) · Zbl 0848.54016
[4] Arhangel’skii, A. V., Normality and dense subspaces, Proc. Amer. Math. Soc., 130, 283-291 (2001) · Zbl 0979.54026
[5] Baturov, D. P., Normality in dense subspaces of products, Topology Appl., 36, 111-116 (1990) · Zbl 0695.54018
[6] Bockstein, M. F., Un théoreme de séparabilité pour les produits topologiques, Fund. Math., 35, 242-246 (1948) · Zbl 0032.19103
[7] van Douwen, E., The Pixley-Roy topology on spaces of subsets, (Set Theoretic Topology (1977), Academic Press: Academic Press New York), 111-134 · Zbl 0372.54006
[8] Engelking, R., General Topology (1989), Helderman: Helderman Berlin · Zbl 0684.54001
[9] W. Just, J. Tartir, A \(κ\); W. Just, J. Tartir, A \(κ\) · Zbl 0907.54011
[10] Kunen, K., Combinatorics, (Barwise, J., Handbook of Mathematical Logic (1977), North-Holland: North-Holland Amsterdam)
[11] Lutzer, D. J., Pixley-Roy topology, Topology Proc., 3, 1, 139-158 (1978) · Zbl 0435.54007
[12] Schepin, E. V., Real-valued functions and spaces close to normal, Sibirsk. Matem. J., 13, 5, 1182-1196 (1972)
[13] F.D. Tall, Normality versus collectionwise normality, in: K. Kunen, J. Vaughan (Eds.), Handbook of Set Theoretic Topology, North-Holland, Amsterdam, pp. 685-733; F.D. Tall, Normality versus collectionwise normality, in: K. Kunen, J. Vaughan (Eds.), Handbook of Set Theoretic Topology, North-Holland, Amsterdam, pp. 685-733
[14] Watson, W. S., Separation in countably paracompact spaces, Trans. Amer. Math. Soc., 290, 2, 831-842 (1985) · Zbl 0583.54013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.