Prove $\mathbb{Q}\cap [0,1]$ is not compact without using Heine-Borel theorem.
36631
Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?
57969
Prove that $0$ is not an essential singularity (UW Madison Qualifying exam)
92740
prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$
29303
Cartesian product of compact sets is compact
99675
A closed ball in $l^{\infty}$ is not compact
140430
Let $E = \mathbb{Q}\big(\left\{\sqrt[n]{2} \mid n \in \mathbb{N} \setminus \left\{0\right\} \right\} \big)$. Prove $[E: \mathbb{Q}]$ is not finite
65358
Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.
34998
How to prove that $\mathbb{Q}$ ( the rationals) is a countable set
140570
Suppose $E\subseteq\mathbb{R}$ is closed and $K\subseteq\mathbb{R}$ is compact. Show that every open cover of $E\cap K$ has a finite subcover
114575
If $\sum_{i=1}^k \dim F_i > n(k-1)$ then $\cap_{i=1}^k F_i \neq \{0\}$
100134
Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC
125894
Prove that $\mathbb{Q}$ belongs to the Borel $\sigma$-algebra of $\mathbb{R}$
147213
Prove that finite unions of compact sets are compact
132382
Showing a set is nowhere dense in $C([0,1])$
98756
Unit ball in $C[0,1]$ not sequentially compact
88099
Not $\sigma$-compact set without axiom of choice
69046
Proof of $\mathbb{Q}$ is not cyclic