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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?2012-06-22
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    The density character of a topological space $(X,\tau)$ is $\min\{|D|: D\subset_{dense} X\}$.2012-06-22

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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?2012-06-22
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    @Pedro: It’s true in all metrizable spaces, because in metrizable spaces the hereditary density equals the density.2012-06-22
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    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!2015-01-29
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    @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](http://en.wikipedia.org/wiki/Cardinal_function#References).2015-01-31
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    @Pedro: My pleasure!2015-02-02