Prove that $\{(x,y)\mid x\in\mathbb Q\}\cup\{(x,y)\mid y\in\mathbb Q\}$ is a connected subset of $\mathbb R ^2.$
15865
Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected.
150738
Proof of $(A - B) - C = A - (B \cup C)$
45337
Suppose that $V, W$ are closed in $X$ and that $V\cap W, V\cup W$ are connected. Show that $V$ and $W$ are connected.
56399
Proof for $A \cup B = B$ if and only if $A \subset B$
27253
Path connectedness of the set $\{(x,y):(x+1)^{2}+y^{2}\leq 1\}\cup\{(x,y):y=x\sin(\frac{1}{x}),x>0\}$
155794
Are the complements of two homeomorphic compact, connected subsets of $\mathbb{R}^2$ homeomorphic?
46106
Prove that $D_r \circ D_s = \{ (x,y) \in \mathbb{R}^2 \ | \ |x-y|<r+s \}$, where $D_a = \{ (x,y) \in \mathbb{R}^2\ | \ |x-y| < a \}$
75604
Proof that $\partial(A\times B) = ((\partial A)\times B)\cup (A\times(\partial B))$
140440
Proving $(A\times C)\cup(B\times D)\subseteq(A\cup B)\times(C\cup D)$
19754
$\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected?
52565
Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.
26302
Prove that $(X \cap Y) \cup Z = (X \cup Z) \cap (Y \cup Z)$
61103
Prove that $\mathbb{|Q| = |Q\times Q|}$
34693
Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
157218
If $E\subseteq X$ is connected, $E\cap A\neq\emptyset$ and $E\cap(X\setminus A)\neq\emptyset$ then $E\cap\partial A\neq\emptyset$
47680
Prove that the union of the interior of a set and the boundary of the set is the closure of the set
36878
Prove that $\operatorname{SL}(n,\Bbb R)$ is connected.