Question

Compare these definitions of limits of a function giving some references about limits of functions between topological spaces.

score 99 · Answer 1 · 2022-07-13 16:19:35

99

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Compare these definitions of limits of a function giving some references about limits of functions between topological spaces.

Published on 13 Jul 2022 - 16:19

score 38 · Answer 2 · 2022-07-20 09:57:58

38

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Let $\alpha := \liminf_{x \to a} f(x)$. Is there a net $(x_t) \subset X$ such that $x_t \to a$ and $f(x_t) \to \alpha$?

Published on 20 Jul 2022 - 9:57

score 70 · Answer 3 · 2022-07-21 01:08:47

70

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Define $g(a) := \liminf_{x \to a} f(x)$ for all $a \in X$. Then $g$ is lower semi-continuous

Published on 21 Jul 2022 - 1:08

score 96 · Answer 4 · 2022-08-08 22:04:58

96

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Definitions of relative countable compactness

Published on 08 Aug 2022 - 22:04

score 49 · Answer 5 · 2026-03-26 07:58:59

49

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12 Possible definitions of basis and their properties

Published on 26 Mar 2026 - 7:58

score 73 · Answer 6 · 2022-10-28 21:33:27

73

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$A$ closed and $B$ compact $\implies \exists U$ open convex and balanced with $(A + U) \cap (B + U) = \emptyset$

Published on 28 Oct 2022 - 21:33

score 213 · Answer 7 · 2026-03-23 08:31:49

213

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Definition of upper semi-continuous functions: limsup or liminf?

Published on 23 Mar 2026 - 8:31

score 90 · Answer 8 · 2022-11-13 18:43:06

90

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Finding a net that converges to the zero map in $\mathbb{R}^\mathbb{R}$

Published on 13 Nov 2022 - 18:43

score 66 · Answer 9 · 2022-11-23 22:40:08

66

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show if convergent nets get mapped to convergent nets then $f:X \rightarrow Y$ is continuous.

Published on 23 Nov 2022 - 22:40

score 55 · Answer 10 · 2022-11-28 14:43:00

55

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Net convergence in initial topology

Published on 28 Nov 2022 - 14:43

score 48 · Answer 11 · 2022-12-15 06:54:02

48

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Cofinality of a set

Published on 15 Dec 2022 - 6:54

score 109 · Answer 12 · 2023-01-05 23:26:31

109

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If $Σ$ is endowed with the initial topology $\cal T_ Φ$ corresponding to the collection $Φ$ then $σ_i⟶σ$ iff $φ(σ_i)⟶φ(σ)$ for any $φ∈Φ$.

Published on 05 Jan 2023 - 23:26

List Question

Compare these definitions of limits of a function giving some references about limits of functions between topological spaces.

Let $\alpha := \liminf_{x \to a} f(x)$. Is there a net $(x_t) \subset X$ such that $x_t \to a$ and $f(x_t) \to \alpha$?

Define $g(a) := \liminf_{x \to a} f(x)$ for all $a \in X$. Then $g$ is lower semi-continuous

Definitions of relative countable compactness

12 Possible definitions of basis and their properties

$A$ closed and $B$ compact $\implies \exists U$ open convex and balanced with $(A + U) \cap (B + U) = \emptyset$

Definition of upper semi-continuous functions: limsup or liminf?

Finding a net that converges to the zero map in $\mathbb{R}^\mathbb{R}$

show if convergent nets get mapped to convergent nets then $f:X \rightarrow Y$ is continuous.

Net convergence in initial topology

Cofinality of a set

If $Σ$ is endowed with the initial topology $\cal T_ Φ$ corresponding to the collection $Φ$ then $σ_i⟶σ$ iff $φ(σ_i)⟶φ(σ)$ for any $φ∈Φ$.

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