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Let $(X,\tau)$ be a topological space. Suppose $dc(X)=\kappa$ and let $D\subset_{dense} X$ be a dense subset of $X$ of cardinality $\kappa$. Is it true that $X\setminus D$ has density character $\kappa$, as a subspace of $X$ with the restricted topology?

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Can you provide a link on which the notion of density character is defined? (or give the definition) Is it the smallest cardinality of a dense subspace? – Davide Giraudo Jun 22 '12 at 15:23
The density character of a topological space $(X,\tau)$ is $\min\{|D|: D\subset_{dense} X\}$. – Pedro Z. Jun 22 '12 at 15:26
up vote 3 down vote accepted

Not necessarily. Let $X=\beta\omega$: $X$ is separable, with $\omega$ as dense subset, but $X\setminus\omega$ is not. An even easier example is a Mrówka $\Psi$-space. Let $\mathscr{A}$ be a maximal almost disjoint family of subsets of $\omega$, and let $X=\omega\cup\mathscr{A}$ with the following topology: points of $\omega$ are isolated, and basic open nbhds of a point $A\in\mathscr{A}$ are of the form $\{A\}\cup(A\setminus F)$ for finite subsets $F$ of $A$. $X$ is separable, since $\omega$ is dense in $X$, but $X\setminus\omega=\mathscr{A}$ is an uncountable discrete set.

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Is this question true in complete metric spaces? – Pedro Z. Jun 22 '12 at 15:42
@Pedro: It’s true in all metrizable spaces, because in metrizable spaces the hereditary density equals the density. – Brian M. Scott Jun 22 '12 at 15:50
Thanks a lot! But I still have a couple of questions. Where can I find the definition of being "hereditary density"? Also, Where can I find the exact result which guarantees that hereditary density implies the result which I am asking? Thanks! – Pedro Z. Jan 29 '15 at 3:38
@Pedro: Two books by István Juhász, Cardinal functions in topology and Cardinal functions in topology — ten years later, are the place to start for information on cardinal functions in topology. Both are freely available as PDFs; you’ll find links in the References for this Wikipedia article. – Brian M. Scott Jan 31 '15 at 22:57
Thank you very much! – Pedro Z. Feb 2 '15 at 0:51

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