View source for Nowhere-dense set

{{MSC|54A05|54C05}}
[[Category:Topology]]
{{TEX|done}}

A subset $A$ of topological space $X$ is nowhere dense if, for every nonempty open $U\subset X$, the intersection $U\cap A$ is ''not'' dense in $U$. Common equivalent definitions are:
* For every nonempty open set $U\subset X$, the interior of $U\setminus A$ is not empty. 
* The closure of $A$ has empty interior.
* The complement of the closure of $A$ is dense.

In a product $X = \prod_\alpha X_\alpha$ of topological spaces, if infinitely many factors are non compact, then any compact subset of $X$ is nowhere dense.

The [[Baire theorem|Baire Category theorem]] asserts that if $X$ is a complete metric space or a locally compact Hausdorff space, then the complement of a countable union of nowhere dense sets is always nonempty.

====References====
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|valign="top"|{{Ref|AP}}|| A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)
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|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,   "Measure and category" , Springer  (1971) {{MR|0393403}} {{ZBL| 0217.09201}}
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|valign="top"|{{Ref|Ke}}||  J.L. Kelley,   "General topology" , v. Nostrand  (1955) {{MR|0070144}} {{ZBL|0066.1660}} 
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