Question

Let $(G_k)$ a sequence of open and dense subsets. Show that $\bigcap_{k\geqslant 0}G_k$ is also open and dense

score 94 · Answer 1 · 2019-10-27 01:59:20

94

Views

Let $(G_k)$ a sequence of open and dense subsets. Show that $\bigcap_{k\geqslant 0}G_k$ is also open and dense

Published on 27 Oct 2019 - 1:59

score 163 · Answer 2 · 2019-10-31 13:13:25

163

Views

On showing that $\{x\in \mathbb{R}\mid\,\limsup_{y\to x} |f'(y)|<\infty\}$ is an open dense subset of $\mathbb{R}$

Published on 31 Oct 2019 - 13:13

score 348 · Answer 3 · 2019-11-01 07:42:40

348

Views

If a set has a second category subset then second category

Published on 01 Nov 2019 - 7:42

score 136 · Answer 4 · 2026-03-25 20:24:11

136

Views

Baire space has isolated point

Published on 25 Mar 2026 - 20:24

score 236 · Answer 5 · 2019-11-04 14:14:19

236

Views

On "simplifying" the definition of a Baire space

Published on 04 Nov 2019 - 14:14

score 1.4k · Answer 6 · 2026-03-27 18:01:02

1.4k

Views

Q(set of rationals) is of baire's first category in itself but N(natural numbers) are of second category in itself .

Published on 27 Mar 2026 - 18:01

score 48 · Answer 7 · 2019-11-13 01:59:02

48

Views

Second category and categorically dense

Published on 13 Nov 2019 - 1:59

score 112 · Answer 8 · 2026-03-26 19:05:06

112

Views

Prove that $\ker(\lambda Id-T)$ stabilizes for some $n\in \Bbb N$

Published on 26 Mar 2026 - 19:05

score 92 · Answer 9 · 2019-12-04 13:22:40

92

Views

Prove that if F closed and of first category in a complete metric space, then F is nowhere dense.

Published on 04 Dec 2019 - 13:22

score 35 · Answer 10 · 2019-12-04 18:18:02

35

Views

Doubt related to Baire Category Theorem

Published on 04 Dec 2019 - 18:18

score 135 · Answer 11 · 2019-12-05 16:05:36

135

Views

Contradiction to Baire's theorem with $\{f\in L^2([0,1]):\int_0^1|f_j|^2\leq j\}$?

Published on 05 Dec 2019 - 16:05

score 69 · Answer 12 · 2019-12-07 20:26:43

69

Views

Is the space $\mathbb{N}^{\mathbb{N}}$ complete and what is its category?

Published on 07 Dec 2019 - 20:26

score 365 · Answer 13 · 2019-12-08 07:59:09

365

Views

Condensation of Singularities, Principle of Uniform Boundedness

Published on 08 Dec 2019 - 7:59

score 262 · Answer 14 · 2019-12-08 15:16:31

262

Views

For a differentiable function $f$ show that $\{x:\limsup_{y\to x}|f'(y)|<\infty\} $ is open and dense in $\mathbb R$

Published on 08 Dec 2019 - 15:16

score 146 · Answer 15 · 2019-12-26 19:16:19

146

Views

Free Ultrafilter limits

Published on 26 Dec 2019 - 19:16

List Question

Let $(G_k)$ a sequence of open and dense subsets. Show that $\bigcap_{k\geqslant 0}G_k$ is also open and dense

On showing that $\{x\in \mathbb{R}\mid\,\limsup_{y\to x} |f'(y)|<\infty\}$ is an open dense subset of $\mathbb{R}$

If a set has a second category subset then second category

Baire space has isolated point

On "simplifying" the definition of a Baire space

Q(set of rationals) is of baire's first category in itself but N(natural numbers) are of second category in itself .

Second category and categorically dense

Prove that $\ker(\lambda Id-T)$ stabilizes for some $n\in \Bbb N$

Prove that if F closed and of first category in a complete metric space, then F is nowhere dense.

Doubt related to Baire Category Theorem

Contradiction to Baire's theorem with $\{f\in L^2([0,1]):\int_0^1|f_j|^2\leq j\}$?

Is the space $\mathbb{N}^{\mathbb{N}}$ complete and what is its category?

Condensation of Singularities, Principle of Uniform Boundedness

For a differentiable function $f$ show that $\{x:\limsup_{y\to x}|f'(y)|<\infty\} $ is open and dense in $\mathbb R$

Free Ultrafilter limits

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