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Hint on Kuratowski 14-Set Theorem Proof

For any set $U$ we have $X/clU=int(X/U)$. Then for $\, U=B^{c}$ we get $X/cl(B^{c})=int(X/B^{c})=intB$. So all we have to prove is that, $cl(intB)=B$. Clearly $\,\,cl(intB)\subseteq\,B$. Now we will ...
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Hint on Kuratowski 14-Set Theorem Proof

The result is true. Since $B$ is the closure of an open set, we have $B = Cl(I(X))$ for some $X$, where $I()$ denotes the interior of a set. So this amounts to showing: $Cl\circ C\circ Cl\circ C\circ ...
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1 vote

Hint on Kuratowski 14-Set Theorem Proof

Notice that $U \subseteq B$ is an open set if and only if $U^c \supseteq B^c$ is closed. So we have \begin{equation}\operatorname{Cl}(B^c)^c = \left(\bigcap_{\substack{F: F \text{ is closed,}} \\ \...
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Question about showing existence

Proof 2 does assume where it says "any basis such that $T v_1 = w_1$" that a basis with that requirement does exist. The reason is exactly the one you used, that we can extend from one non-...
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