"Any open subset of the reals is a countable union of disjoint open intervals". Do we need the axiom of choice to prove this?
127575
Prove that Open Sets in $\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice
44522
Subsets of the reals when the Continuum Hypothesis is assumed false
24372
Infinite Set is Disjoint Union of Two Infinite Sets
4518
How would one go about proving that the rationals are not the countable intersection of open sets?
83757
Is the following equivalent to the axiom of choice?
33563
Is the axiom of choice used in the proof that every open set is the union of basis elements (Mukres, Lemma 13.1)
132046
Looking for Cantor's original proof of the Cantor-Bernstein theorem that relies on the axiom of choice?
16654
Unions and Intersections of Open Sets are Open
7607
How to prove that $\mathbb{Q}$ is not the intersection of a countable collection of open sets.
97697
Countable union of sets of cardinality $c$ has cardinality $c$
128730
Open subspaces of locally compact Hausdorff spaces are locally compact
94825
Open set as a finite union of closed sets
66301
The cardinality of the set of countably infinite subsets of an infinite set
33157
Proof that an infinite subset of $\mathbb{N}$ is countable
23442
Cardinality of the union of infinite and countable sets
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.
114608
Proof concerning regular space: there exist a closed set contained in any open set containing $x$