query: Let (X, d) be a metric space and let A be a subset of X and O be an open subset of X. Prove that $O \cap \overline{A} \subseteq \overline{O\cap A}$
positive_passages.docid: 152193
positive_passages.text: The intersection of a closure with an open set
negative_passages.docid: 134770
negative_passages.text: For sets $A,B$, $A \cap (B \setminus A) \subseteq \varnothing$
negative_passages.docid: 118332
negative_passages.text: proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
negative_passages.docid: 88185
negative_passages.text: Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed
negative_passages.docid: 157218
negative_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: 154978
negative_passages.text: Some equivalent formulations of compactness of a metric space
negative_passages.docid: 55016
negative_passages.text: Prove that if ($d(x,y)=0$ iff $x=y$) and if $d(x,z)\leq d(x,y)+d(z,y)$ then $d$ is a metric
negative_passages.docid: 137707
negative_passages.text: How to prove that $(A\cup B) \cap C \subseteq A\cup (B\cap C)$
negative_passages.docid: 151984
negative_passages.text: Proving the existence of a metric space
negative_passages.docid: 51227
negative_passages.text: Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$
negative_passages.docid: 15480
negative_passages.text: Open set in a metric space is union of closed sets
negative_passages.docid: 123753
negative_passages.text: In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
negative_passages.docid: 143735
negative_passages.text: Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$
negative_passages.docid: 64260
negative_passages.text: Prove that a function between metric spaces is continuous iff the inverse image of a closed set is closed
negative_passages.docid: 28391
negative_passages.text: Prove the compactness of a set B
negative_passages.docid: 157753
negative_passages.text: Show that any set in a metric space can be written as the intersection of open sets
negative_passages.docid: 135315
negative_passages.text: Proving whether a function on the set of real sequences is a metric