Show that exists $U$ and $V$ such that $x\in V$, $K\subset U$ and $U\cap V=\emptyset$
102168
Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
37183
Show that the set of all $x$ such that $x \in P$ and $x \notin Q$ exists
141529
Proving existence of a minimum
858
Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?
74623
Associative rule of subspaces and vector spaces: $(U+V) \cap W = U + (V \cap W)$
27107
intersection of the empty set and vacuous truth
47680
Prove that the union of the interior of a set and the boundary of the set is the closure of the set
34099
Proving that the empty set is unique
130156
Image of the union and intersection of sets.
54737
Proof that $V = \sum_k U_k$ and $\dim V=\sum_k \dim U_k$ implies $V=\oplus_k U_k$
133152
(X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀𝑥 ∈ X, B(x,ε) is contained in some member of the cover.
49143
Showing $A \subset B \iff A\cap B=A$
23225
Union of connected subsets is connected if intersection is nonempty
77791
Proving an inclusion related to algebraic sets and interpreting it
15560
Prove that the union of three subspaces of $V$ is a subspace iff one of the subspaces contains the other two.
142229
Prove that if $V \cap \text{Span}\{u_1, u_2,...,u_k\} = \{ 0\} $, then $Au_1, Au_2,...,Au_k$ are linearly independent
60707
Show that if $ \ A \cup B = A$ and $ \ A\cap B = A$ then $ \ A = B$