Question

Lower semi-continuous function attains minimum on a compact set

score 743 · Answer 1 · 2022-02-23 14:35:08

743

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Lower semi-continuous function attains minimum on a compact set

Published on 23 Feb 2022 - 14:35

score 127 · Answer 2 · 2022-02-23 17:36:33

127

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Characterize lower semi-continuous functions by net convergence

Published on 23 Feb 2022 - 17:36

score 28 · Answer 3 · 2022-02-24 12:24:42

28

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Let $(x_d), (y_d)$ be nets such that $x_d \to a$ and $y_d \in \overline{\operatorname{conv} \{x_e \mid e \ge d\}}$. Then $y_d \to a$

Published on 24 Feb 2022 - 12:24

score 54 · Answer 4 · 2022-03-10 08:10:01

54

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Showing that a mapping $f:X \to Y$ is continuous at $z_0 \in X$ when the net $f \circ \varphi$ converges to $f(z_0)$ with $\varphi\to z_0$

Published on 10 Mar 2022 - 8:10

score 96 · Answer 5 · 2022-03-19 02:23:57

96

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Countable (Willard) subnet in first-countable space

Published on 19 Mar 2022 - 2:23

score 80 · Answer 6 · 2026-03-26 17:51:22

80

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Banach limit for nets

Published on 26 Mar 2026 - 17:51

score 116 · Answer 7 · 2026-02-24 05:07:46

116

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Does every net has a convergent subnet in a compact uniform space?

Published on 24 Feb 2026 - 5:07

score 44 · Answer 8 · 2022-04-20 19:34:02

44

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Let $E$ be a t.v.s. and $f$ a discontinuous linear functional on $E$. There is a net $(x_d)$ such that $x_d \to 0$ and $f(x_d) = 1$ for all $d$

Published on 20 Apr 2022 - 19:34

score 129 · Answer 9 · 2022-05-04 13:40:22

Exercise $11$.D.c from Willard: «Prove that if every subnet of $(x_λ)_{λ\inΛ}$ has a subnet converging to $x$ then $(x_λ)_{λ\inΛ}$ converges to $x$»

score 24 · Answer 10 · 2022-05-12 17:04:49

24

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How to prove that every Cauchy net in a complete metric space is convergent?

Published on 12 May 2022 - 17:04

score 160 · Answer 11 · 2026-03-26 02:44:45

160

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If $X$ is first-countable then a net converges when a subsequence converges?

Published on 26 Mar 2026 - 2:44

score 135 · Answer 12 · 2022-06-05 00:26:15

135

Views

Which are more standard: Filterbases or nets? What are the benefits of each?

Published on 05 Jun 2022 - 0:26

score 91 · Answer 13 · 2022-06-06 18:32:02

91

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Let $A$ and $B$ be nonempty disjoint closed sets such that $A$ is compact. There exists a neighborhood $V$ of $0$ such that $(A+V) \cap B=\emptyset$

Published on 06 Jun 2022 - 18:32

score 200 · Answer 14 · 2022-06-12 08:13:47

200

Views

Prove or disprove if two continuous functions are equal when they agree in a dense set.

Published on 12 Jun 2022 - 8:13

score 184 · Answer 15 · 2022-07-04 19:10:07

184

Views

Is a topology determined by the convergence of nets?

Published on 04 Jul 2022 - 19:10

List Question

Lower semi-continuous function attains minimum on a compact set

Characterize lower semi-continuous functions by net convergence

Let $(x_d), (y_d)$ be nets such that $x_d \to a$ and $y_d \in \overline{\operatorname{conv} \{x_e \mid e \ge d\}}$. Then $y_d \to a$

Showing that a mapping $f:X \to Y$ is continuous at $z_0 \in X$ when the net $f \circ \varphi$ converges to $f(z_0)$ with $\varphi\to z_0$

Countable (Willard) subnet in first-countable space

Banach limit for nets

Does every net has a convergent subnet in a compact uniform space?

Let $E$ be a t.v.s. and $f$ a discontinuous linear functional on $E$. There is a net $(x_d)$ such that $x_d \to 0$ and $f(x_d) = 1$ for all $d$

Exercise $11$.D.c from Willard: «Prove that if every subnet of $(x_λ)_{λ\inΛ}$ has a subnet converging to $x$ then $(x_λ)_{λ\inΛ}$ converges to $x$»

How to prove that every Cauchy net in a complete metric space is convergent?

If $X$ is first-countable then a net converges when a subsequence converges?

Which are more standard: Filterbases or nets? What are the benefits of each?

Let $A$ and $B$ be nonempty disjoint closed sets such that $A$ is compact. There exists a neighborhood $V$ of $0$ such that $(A+V) \cap B=\emptyset$

Prove or disprove if two continuous functions are equal when they agree in a dense set.

Is a topology determined by the convergence of nets?

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