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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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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: $Cl\circ C\circ Cl\circ C\circ ... • 2,934 1 vote ### Hint on Kuratowski 14-Set Theorem Proof Notice that$U \subseteq B$is an open set if and only if$U^c \supseteq B^c$is closed. So we have \operatorname{Cl}(B^c)^c = \left(\bigcap_{\substack{F: F \text{ is closed,}} \\ \... • 4,091 1 vote Accepted ### Question about showing existence Proof 2 does assume where it says "any basis such that$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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