query: if one of the sets A and B is compact then d(A,B)>0.
positive_passages.docid: 2978
positive_passages.text: A and B disjoint, A compact, and B closed implies there is positive distance between both sets
negative_passages.docid: 8015
negative_passages.text: Compact set plus a closed set is a closed set
negative_passages.docid: 67259
negative_passages.text: Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
negative_passages.docid: 95294
negative_passages.text: Dense subset of $C[0,1]$ which preserves rationals
negative_passages.docid: 106534
negative_passages.text: "Closed subsets of compact space are compact" with an alternative definition of covering
negative_passages.docid: 148994
negative_passages.text: $K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
negative_passages.docid: 36631
negative_passages.text: Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?
negative_passages.docid: 153670
negative_passages.text: Prove that if $a \mid b$ and $b \mid c$, then $a \mid c$.
negative_passages.docid: 10909
negative_passages.text: $|A \cup B| = \mathfrak{c}$, then $A$ or $B$ has cardinality $\mathfrak{c}$.
negative_passages.docid: 75199
negative_passages.text: $a \equiv b \pmod n$ and $c\equiv d \pmod n$ implies $ac \equiv bd \pmod n$
negative_passages.docid: 134914
negative_passages.text: Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$.
negative_passages.docid: 31559
negative_passages.text: If $Z\subseteq Y\subseteq X$ and $Z$ is compact in $Y$, then $Z$ is compact in $X$
negative_passages.docid: 44076
negative_passages.text: Proving if $d_0$ is the smallest positive integer in $S$ then $d_0 = \gcd(a,b)$
negative_passages.docid: 19566
negative_passages.text: Continuous images of compact sets are compact
negative_passages.docid: 134770
negative_passages.text: For sets $A,B$, $A \cap (B \setminus A) \subseteq \varnothing$
negative_passages.docid: 146146
negative_passages.text: Proof that a closed subset of a compact space is a compact space
negative_passages.docid: 77482
negative_passages.text: Continuous and preserves measurability $\implies$ preserves null sets.