Tagged Questions
4
votes
0answers
56 views
Partition of $\mathbb{R}$ into nullset and 1st category set
Let $\{a_i\}_{i=1}^{\infty}$ be an enumeration of the rationals, $\mathbb{Q}$.
Let $I_{ij}$ be the open interval centered at $a_i$ and having length $1/2^{i+j}$, and define $G_j = \cup_{i=1}^\infty ...
4
votes
4answers
203 views
(ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior
Since the specific space $\mathbb{R}^k$ is given, this might be provable in ZF.
Let $\{F_n\}_{n\in \omega}$ be a family of closed subset of $\mathbb{R}^k$, of which the union is $\mathbb{R}^k$.
...
85
votes
1answer
2k views
Does the open mapping theorem imply the Baire category theorem?
A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice.
On the other hand, the three ...
