I recently came across the following remark in a book: "Notice that a constructible set contains a dense open subset of its closure." Now this doesn't seem at all obvious to me.(adsbygoogle = window.adsbygoogle || []).push({});

Let us recall the definitions first. A locally closed set is the intersection of a closed and an open set. A constructible set is a finite union of locally closed sets.

Let [tex]H[/tex] denote our constructible set in a topological space [tex]X[/tex]. Let [tex]H^c[/tex], [tex]H^{\circ}[/tex], [tex]H'[/tex] denote closure, interior and complement respectively.

Now [tex]H^{\circ} \subseteq H \subseteq H^c[/tex] is open so this seems like an obvious choice. We just need to show that it is dense in the latter. Well any set is dense inside its closure so it would be sufficient to show that [tex](H^{\circ})^c = H^c[/tex].

Note that [tex]LHS \subseteq RHS[/tex] is obvious.

The following facts follow from the definitions:

1) The closure of a union is the union of the closures.

2) The closure of an intersection is a subset of the intersection of the closures.

3) The interior of an intersection is the intersection of the interiors.

4) The interior of a union is a subset of the union of the interiors.

So let [tex]H = (O_1 \cap C_1) \cup \dots \cup (O_n \cap C_n)[/tex] for [tex]C_i, O_i[/tex] closed and open respectively.

[tex]H^c = (O_1 \cap C_1)^c \cup \dots \cup (O_n \cap C_n)^c \subseteq (O_1^c \cap C_1) \cup \dots \cup (O_n^c \cap C_n)\ \ \ (*)[/tex]

On the other hand

[tex]H^{\circ} \supseteq (O_1 \cap C_1)^{\circ} \cup \dots \cup (O_n \cap C_n)^{\circ} = (O_1 \cap C_1^{\circ}) \cup \dots \cup (O_n \cap C_n^{\circ})[/tex]

hence

[tex](H^{\circ})^c \supseteq (O_1 \cap C_1^{\circ})^c \cup \dots \cup (O_n \cap C_n^{\circ})^c[/tex].

If this were to equal (*), we'd be done. A sufficient condition would seem to be that

[tex] (U_1 \cap U_2)^c = U_1^c \cap U_2^c [/tex] for any [tex] U_1, U_2 [/tex] open.

I've tried this but I can't quite nail it.

Am I on the right track or am I making life difficult for myself? Thanks in advance!

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# Constructible sets contain a dense open subset of their closure

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