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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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A property of subspaces of a topological space

Let $X$ be a topological space and $A$ be a subset of $X$. Is there any nice condition on $A$ equivalent to the property that if $C$ is a connected component of $X$, then either $A\cap C=\emptyset$ or …
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