query: Prove if metric space $X$ contains a connected dense subset, then $X$ is connected.
positive_passages.docid: 18039
positive_passages.text: Let $A$ be a dense subset of $X$ and suppose $A$ is connected in the induced subspace topology, then $X$ is also connected
negative_passages.docid: 102598
negative_passages.text: Show that $X$ is path connected if and only if each $X_\alpha$ is path connected.
negative_passages.docid: 3649
negative_passages.text: Closed unit interval is connected proof
negative_passages.docid: 153256
negative_passages.text: Compact topological space with closed graph implies continuity
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: 12104
negative_passages.text: How to prove this result involving the quotient maps and connectedness?
negative_passages.docid: 107701
negative_passages.text: Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$
negative_passages.docid: 153492
negative_passages.text: Let $p\colon X \to Y$ be a quotient map then if each set $p^{-1}({y})$ is connected, and if $Y$ is connected, then $X$ is connected.
negative_passages.docid: 26563
negative_passages.text: Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
negative_passages.docid: 108619
negative_passages.text: Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty
negative_passages.docid: 42811
negative_passages.text: Product of path connected spaces is path connected
negative_passages.docid: 59654
negative_passages.text: Show that in a discrete metric space, every subset is both open and closed.
negative_passages.docid: 90590
negative_passages.text: Prove that the following set is dense
negative_passages.docid: 83163
negative_passages.text: If $X$ is a connected metric space, then a locally constant function $f: X \to $ M, $M $ a metric space, is constant
negative_passages.docid: 158189
negative_passages.text: If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected
negative_passages.docid: 44430
negative_passages.text: Is there an open dense set $S \subset [0,1]$ such that $m(S)<1$?
negative_passages.docid: 99645
negative_passages.text: Metrizable topological space $X$ with every admissible metric complete then $X$ is compact