$f:X\to Y$ continuous and open, prove: if $D\subseteq Y$ is dense $\implies f^{-1}[D]$ is dense
67602
The inverse image of dense set is dense and of a comeager set is comeager?.
128069
Continuity in Slices + Mapping Compact Sets to Compact Sets = Continuous
142459
Continuity of a Function in Terms of Closure and Interior
18039
Let $A$ be a dense subset of $X$ and suppose $A$ is connected in the induced subspace topology, then $X$ is also connected
88523
Show that $A=\{ \frac{m}{2^n}:m\in \mathbb {Z},n\in \mathbb {N} \} $ is dense in $\mathbb {R}$
30568
Let $q \in \mathbb C$, $\|q\|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n: n \in \mathbb N\}$ is dense in $S^1$
99179
Continuous function with compact domain has continuous inverse
106434
Prove that for a dense set $A$ and an open set $U$, $U\subset \overline{A\cap U}$
130975
Given a pair of continuous functions from a topological space to an ordered set, how to prove that this set is closed?
84512
The preimage of continuous function on a closed set is closed.
145368
How to show $f:Y\to X^{+}$ is continuous?
52388
The equivalence of open-set definition of continuous mappings to the $\epsilon-\delta$ definition
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
24083
$f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
130299
Prove that if a function $f: X\to Y$ continuous then its graph is closed
155923
continuity of $f(x,y)$ in $\mathbb R^2$.