Set Closure Preserves Set Inclusion
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $B \subseteq A \subseteq S$.
Then:
- $B^- \subseteq A^-$
where $A^-$ and $B^-$ are the set closures in $T$ of $A$ and $B$ respectively.
Proof
By definition 1 of set closure:
- $B^- = B \cup B'$
where $B'$ is the derived set of $B$ in $T$.
Similarly:
- $A^- = A \cup A'$
where $A'$ is the derived set of $A$ in $T$.
From Derived Set Preserves Set Inclusion:
- $B' \subseteq A'$
So, by Set Union Preserves Subsets:
- $B \cup B' \subseteq A \cup A'$
That is:
- $B^- \subseteq A^-$
$\blacksquare$