Prove the union $\bigcup A_n$ has empty interior in $X$
14215
Show that the union $\bigcup A_n$ has empty interior in $X$
93952
Show that $X$ can be represented as a union of disjoint equivalence classes
78463
Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$
117781
Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition
143003
$\bigcup \emptyset$ is defined but $\bigcap \emptyset$ is not. Why?
11834
Prove that $\liminf x_n = -\limsup (-x_n)$
48593
Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$
138069
Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
10107
Prove $A\cup (A\cap B) = A$
127575
Prove that Open Sets in $\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice
80212
Suppose that $X$ is a topological space and $x_0\in X$. Prove that $\widetilde{H_n}(X)=H_n(X,x_0)$ for all $n\geq 0$.
47680
Prove that the union of the interior of a set and the boundary of the set is the closure of the set
136755
Direct proof of empty set being subset of every set
127900
Interior of closure of open set in topological space
84437
Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$
56399
Proof for $A \cup B = B$ if and only if $A \subset B$
113377
Empty preimage of an intersection implies empty intersection of the preimages