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Homework Statement
Let [tex]X[/tex] be a locally compact, Hausdorff topological space. If [tex]x[/tex] is an element of [tex]X[/tex] and [tex]U[/tex] is a neighborhood of [tex]x[/tex], find a compact neighborhood of [tex]x[/tex] contained in [tex]U[/tex].
Homework Equations
The Attempt at a Solution
Let [tex]N[/tex] be a compact neighborhood of [tex]x_[/tex]. The set [tex]D=Fr(N\cap\bar U)[/tex] is closed, hence compact. For each [tex]y\in D[/tex], there exist disjoint neighborhoods [tex]N_y[/tex] and [tex]N_y'[/tex] of [tex]y[/tex] and [tex]x[/tex], respectively. The set [tex]\{N_y:y\in D\}[/tex] is an open cover of [tex]D[/tex], hence it has a finite subcover [tex]\{N_{y_n}:y_n\in D\}[/tex]. The set [tex]\cap \bar N_{y_n}'}\subset N\cap U[/tex] is a closed neighborhood of [tex]x[/tex], hence it is compact.
Is this correct?
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