Prove that $f(A)$ is a separable subset
53859
Continuous image of separable space
58948
If $A$ is compact, is then $f(A)$ compact?
64320
Prove that if $0<a<Pr(A)$ then there exists $B\subset A$ such that $Pr(B)=a$
140095
Let $k^p = k$, $K = k(x)$ and $F = k(u)$ for some $u \in K$. Show that $K/F$ is separable $\iff u \not\in K^p$.
17212
The preimage of a subset
28391
Prove the compactness of a set B
83311
Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}
68918
Show that $A\cap B\subseteq A$ and $A\subseteq A\cup B$
122271
A subset of a transitive set is transitive
60348
How do I prove $F(a)=F(a^2)?$
78463
Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$
53627
Proving continuity of $f$
137707
How to prove that $(A\cup B) \cap C \subseteq A\cup (B\cap C)$
49592
Show that, if $f:A\to B$ is a function, with $A$ and $B$ being finite sets, and $|A|=|B|$, then $f$ is one to one iff $f$ is onto.
70264
Prove that the inverse image of an open set is open
118332
proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
83009
Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field