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?
Density character of a subspace of a topological space.
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general-topology
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0The density character of a topological space $(X,\tau)$ is $\min\{|D|: D\subset_{dense} X\}$. – 2012-06-22
1 Answers
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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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0@Pedro: My pleasure! – 2015-02-02