real analysis - Why is $\ell^\infty (\mathbb {N})$ not separable ...
My functional analysis textbook says "The metric space l∞ l ∞ is not separable." The metric defined between two sequences {a1,a2,a3 …} {a 1, a 2, a 3} and {b1,b2,b3, …} {b 1, b 2, b 3,} …
Prove if $X$ is a compact metric space, then $X$ is separable.
Let X be a metric space. Prove if X is compact, then X is separable. X separable X X contains a countable dense subset. E ⊂ X E⊂X dense in X ¯ E = X X E¯¯¯¯=X . X X compact every open …
Prove that a subspace of a separable and metric space is itself …
Prove that a subspace of a separable and metric space is itself separable Ask Question Asked 12 years, 3 months ago Modified 2 months ago
Definition of Separable Space - Mathematics Stack Exchange
Oct 8, 2020 · The standard definition (e.g. from wikipedia) that a separable topological space X X contains a countable, dense subset, or equivalently that there is a sequence (xn) (x n) of …
Is $L^p$ separable? - Mathematics Stack Exchange
Jun 27, 2014 · Wikipedia en.wikipedia.org/wiki/Separable_space#Non-separable_spaces: The Lebesgue spaces Lp, over a separable measure space, are separable for any 1 ≤ p < ∞.
$X^*$ is separable then $X$ is separable [Proof explanation]
Feb 5, 2020 · $X^*$ is separable then $X$ is separable Proof: Here is my favorite proof, which I think is simpler than both the one suggested by David C. Ullrich and the one I had ...
functional analysis - $C (X)$ is separable when $X$ is compact ...
Jun 19, 2015 · this result is not trivial: If X is a compact T2 T 2 space X X, then C(X) C (X) is separable iff there is a metric X × X → R X × X → R that induces the topology of X X. You …
galois theory - The definition of the separable closure of a field ...
Mar 7, 2024 · Non-separable extensions and elements are not so nice in some ways, in particular recall that an extension is Galois if it is normal and separable. So one might consider only …
general topology - Prove that $\mathbb {R}^k$ is separable ...
I'd like to show that $\\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if
Every subspace of a separable metric space is separable.
Dec 2, 2017 · IIf it were right it would apply to every separable space because you have not used any of the metric properties. But a separable non-metrizable space can have a non-separable …