Let $\{U_\alpha\}$ be a finite open cover for $X$. Then for $\varepsilon > 0$ every $x \in X, B(x,\varepsilon)$ is contained in one of $U_\alpha$’s.
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.
138105
Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$.
152927
seemingly nontrivial question about covering maps and evenly covered open sets
143043
When is a local homeomorphism a covering map?
84021
How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?
153934
How to prove that a set R\Z is open
138039
If $X$ is Hausdorff then every finite set is closed.
73130
A simple question about open set
154559
For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
49220
Open Sets of $\mathbb{R}^1$ and axiom of choice
13162
Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit
21599
$(0,1)$ is an open subset of $\mathbb{R}$ but not of $\mathbb{R}^2$, when we think of $\mathbb{R}$ as the x-axis in $\mathbb{R}^2$. Prove this.
82405
For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $
113696
Showing that the unit ball in $(C^\alpha[a, b], \|\cdot\|_\alpha)$ is compact in $(C[a, b], \|\cdot\|_\infty)$
107370
An open covering of $\mathbb{Q} \cap [0,1]$ that does not contain any finite subcovering
142740
If $\varphi \in E'$ and $A$ is convex and open then $\varphi (A)$ is an open interval
110832
$F_\sigma$-subsets in a normal space can be separated