query: Prove that preimage of any closed subset in $\mathbb{R}$ is closed subset in $\mathbb{R}^{n}$.
positive_passages.docid: 157603
positive_passages.text: Continuity $\iff$ Preimage is closed whenever set is closed
negative_passages.docid: 117478
negative_passages.text: Prove that $\mathbb{R}$ and the interval $(0, \infty)$ have the same cardinality.
negative_passages.docid: 80827
negative_passages.text: Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...
negative_passages.docid: 22464
negative_passages.text: Which of the following subsets of   $\mathbb{R}^2$ are compact
negative_passages.docid: 2198
negative_passages.text: Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous
negative_passages.docid: 46698
negative_passages.text: Showing a set is closed under inverses
negative_passages.docid: 127575
negative_passages.text: Prove that Open Sets in $\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice
negative_passages.docid: 148992
negative_passages.text: A finite set is closed
negative_passages.docid: 10911
negative_passages.text: $A \in M_{n}(\mathbb{R})$ has inverse if and only if it's rank is n.
negative_passages.docid: 138202
negative_passages.text: Every open set in $\mathbb{R}$ is a countable union of closed sets
negative_passages.docid: 95244
negative_passages.text: Prove that a finite union of closed sets is also closed
negative_passages.docid: 46917
negative_passages.text: How to prove that $GL(n,\mathbb R)$ is not connected subset and open subset of$M_n (\mathbb{R})$
negative_passages.docid: 125962
negative_passages.text: The set $A'$ is closed in a $T_1$ space
negative_passages.docid: 52565
negative_passages.text: Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.
negative_passages.docid: 143067
negative_passages.text: Show given any $x\in\mathbb{R}$ show there exists a unique $n\in\mathbb{Z}$ such that $n-1\leq x <n$.
negative_passages.docid: 70576
negative_passages.text: Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
negative_passages.docid: 147914
negative_passages.text: Closed image of locally compact space