# Questions tagged [separable-spaces]

For questions about separable spaces, i.e., topological spaces containing a countable dense set.

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### Seperability for the collections of all non-empty compact subsets of $\mathbb{R}^2$ with Hasudorff metric

Let $X$ be the collections of all non-empty compact subsets of $\mathbb{R}^2$, which has the Euclidean metric. Let $(X,d)$ be a metric space, where $d$ is the Hasudorff metric. Is $X$ separable? ...
3 votes
1 answer
66 views

### Proving a Proposition about separable Hausdorff spaces that are locally euclidean

Let X be a separable Hausdorff space such that for every $x \in X$ there exists an open neighborhood $U$ of $x$ such that $U$ is homeomorphic to an open subset of $\mathbb{R}^n$. Show that: (i) $X$ is ...
• 409
2 votes
2 answers
82 views

• 103
1 vote
1 answer
31 views

### A separable normed space that is continuously embedded in a non-separable normed space implies that this embedding isn't dense.

As a preliminary I introduce the definition of denseness I am using: Definition (dense subsets of metric spaces). Suppose $(M,d)$ is a metric space. A subset $S \subset M$ is called dense in $M$ if ...
• 1,095
1 vote
1 answer
68 views

### Separability of codomains of Borel functions taking values in completely regular spaces

I am looking for a reference (or a counterexample) to the following statement. Let $X$ be a separable metric space. Suppose that $Y$ is a completely regular topological space and $f\colon X\to Y$ is a ...
• 16.4k
3 votes
1 answer
241 views

### Adapting a proof of the non-separability of Morrey Spaces for a different definition.

In the article "Morrey spaces, their duals and preduals", by Marcel Rosenthal and Hans Triebel, for every $1 \leqslant p < \infty$ and $-\frac{n}{p} < r < 0$ the Morrey Spaces are ...
• 1,095
2 votes
2 answers
76 views

### Are dual spaces to separable normed spaces first-countable?

In [1] I found the following theorem (roughly translated by me) Satz 13.10 If $X$ is a separable normed $k$-vector space ($k = \mathbb R$ or $\mathbb C$) with continuous dual space $X'$, then the ...
• 8,605
1 vote
0 answers
35 views

0 votes
1 answer
105 views

### Subspace of Euclidean space

Consider the euclidean space $\mathbb{R}^n$. Consider a closed compact subset $A\subset\mathbb{R}^n$. For example, take $A=[a,b]^n$, with $0<a<b<\infty$. It is well know that $\mathbb{R}^n$ ...
8 votes
2 answers
88 views

### Prove that $X := \{ f: [0,1] \to [0,1] : f \text{ is continuous and } f(1) = 0 \}$ with the given distance is neither connected nor separable

Prove that $X := \{ f: [0,1] \to [0,1] : f \text{ is continuous and } f(1) = 0 \}$ with $d(f,g) := \inf \{r \geq 0 : f(t) = g(t) \forall r ≤ t ≤ 1 \}$ is neither connected nor separable. Here is my ...
• 3,638
1 vote
1 answer
101 views

• 442