query: Let $C \subset X$ be a connected space and $A \subset X$ s.t. $C\cap A \neq \emptyset$ and $C\cap (X-A) \neq \emptyset$. Prove...
positive_passages.docid: 157218
positive_passages.text: If $E\subseteq X$ is connected, $E\cap A\neq\emptyset$ and $E\cap(X\setminus A)\neq\emptyset$ then $E\cap\partial A\neq\emptyset$
negative_passages.docid: 140570
negative_passages.text: Suppose $E\subseteq\mathbb{R}$ is closed and $K\subseteq\mathbb{R}$ is compact. Show that every open cover of $E\cap K$ has a finite subcover
negative_passages.docid: 134703
negative_passages.text: Let $X$ be a connected space, $f:X\to X$ continuous involution, $g:X\to\mathbb{R}$ continuous. Prove that $\exists$ $x\in X$ such that $g(x)=g(f(x))$.
negative_passages.docid: 125962
negative_passages.text: The set $A'$ is closed in a $T_1$ space
negative_passages.docid: 39455
negative_passages.text: Prove that $(X\times Y)\setminus (A\times B)$ is connected
negative_passages.docid: 26919
negative_passages.text: Intersection of Connected Sets
negative_passages.docid: 151719
negative_passages.text: Connected set in $\mathbb{S}^1$
negative_passages.docid: 39293
negative_passages.text: Given a compact set $K$ and closed set $E,$ prove: $E\cap K\neq \emptyset $
negative_passages.docid: 33524
negative_passages.text: Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$
negative_passages.docid: 13415
negative_passages.text: How to show if $A \subseteq B$, then $A \cap B = A$?
negative_passages.docid: 46056
negative_passages.text: If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A
negative_passages.docid: 43310
negative_passages.text: Proving $A\cap(B-C)$ is equal to $(A\cap B)-(A\cap C)$
negative_passages.docid: 13132
negative_passages.text: If the union of two sets is contained in the intersection, then one is contained in the other ($[A\cup B\subseteq A\cap B]\implies A \subseteq B$)
negative_passages.docid: 41477
negative_passages.text: Prove $f(S \cap T) \subseteq f(S) \cap f(T)$
negative_passages.docid: 111008
negative_passages.text: Show that $X $ is a infinite set ,it is connected in the finite complement topology?
negative_passages.docid: 146146
negative_passages.text: Proof that a closed subset of a compact space is a compact space
negative_passages.docid: 137707
negative_passages.text: How to prove that $(A\cup B) \cap C \subseteq A\cup (B\cap C)$