query: connectedness of A={$(x,y):x \in \mathbb{Q}\,$ or$ \, y \in \mathbb{Q}$}
positive_passages.docid: 15865
positive_passages.text: Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected.
negative_passages.docid: 127919
negative_passages.text: prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected.
negative_passages.docid: 116932
negative_passages.text: Proving a simple graph is a connected graph
negative_passages.docid: 124234
negative_passages.text: Prove that if $X$ and it's closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected.
negative_passages.docid: 153107
negative_passages.text: Does every connected component of $\{z : |P(z)|<1 \}$ contain a zero?
negative_passages.docid: 59424
negative_passages.text: Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
negative_passages.docid: 30079
negative_passages.text: Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?
negative_passages.docid: 145039
negative_passages.text: Algebraic independence over $\overline{\mathbb Q}$ and over $\mathbb Q$
negative_passages.docid: 109682
negative_passages.text: Proposed proof of: If $A \subset B \subset \bar{A}$ and $A$ is connected, then $B$ is connected
negative_passages.docid: 85819
negative_passages.text: A non-polyhedral pair $(X,A)$ with both $X$ and $A$ polyhedral
negative_passages.docid: 129109
negative_passages.text: Is the set of polynomial with coefficients on $\mathbb{Q}$ enumerable?
negative_passages.docid: 5132
negative_passages.text: Given $A\cap \overline{B}\neq \emptyset$, prove $A\cup B$ connected.
negative_passages.docid: 17470
negative_passages.text: Does path-connected imply simple path-connected?
negative_passages.docid: 133633
negative_passages.text: A question about path-connected and arcwise-connected spaces
negative_passages.docid: 116494
negative_passages.text: What happens to the connectedness of $\mathbb R^2$ when countable many points are removed?
negative_passages.docid: 139064
negative_passages.text: Show by using logical connectives laws that $(P\to Q) \land (Q \to R) $ is equivalent to $(P \to R) \land [(P \iff Q) \lor (R \iff Q)]$
negative_passages.docid: 18039
negative_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