# Questions tagged [dense-subspaces]

For questions related to dense subspaces. In general topological spaces, a dense set is one whose intersection with any nonempty open set is nonempty.

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### 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 ...
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1 vote
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### Necessity of Hausdorff-ness in "continuous function determined by its values on a dense subset"

It's well-known that if a continuous function taking values in a Hausdorff space is uniquely determined by its specification on a dense subset of the domain. Now, I contemplate on the necessity of ...
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1 vote
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### The complement of any set of first category on the line is dense iff the intersection of any sequence of dense open sets is dense

I want to prove that The complement of any set of first category on the line is dense iff the intersection of any sequence of dense open sets is dense. but I don't need to prove this in general. I ...
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1 vote
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### If two function agree on a dense set, what condition is necessary for them to be equal?

Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$, and $g$ a continuous function. If $f(x)=g(x)$ for every $x$ in a dense set $A$ in $\mathbb{R}$, does $f$ having the Darboux property ...
1 vote
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### Given $\varepsilon > 0$ and an irrational $\alpha$, there exists a rational number $m / n$ that is ($\varepsilon / n$)-close to $\alpha$?

Since $\mathbb{Q}$ is dense in $\mathbb{R}$, given a real $\varepsilon > 0$ and an $\alpha$, there exist $m, n \in \mathbb{Z}$ such that $$\left|\frac{m}{n} - \alpha \right| < \varepsilon \,.$$ ...
1 vote
63 views

### Topology with given infinite dense sets

Suppose $\mathcal{A}$ is a non-empty upward-closed family (i.e. $A\in \mathcal{A}$ and $A\subseteq B$ implies $B\in\mathcal{A}$) of infinite subsets on an infinite set $X$. In ZFC, does there exist ...
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### Topology with given dense sets

Suppose $\mathcal{A}$ is a non-empty upward-closed family (i.e. $A\in \mathcal{A}$ and $A\subseteq B$ implies $B\in\mathcal{A}$) of non-empty subsets on a non-empty set $X$. In ZFC, does there exist ...
• 10k
1 vote
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### intersection of sets not dense implies interior of closure of intersection is not dense

$G_n$ is a countable family of dense open sets X is a metric space I would like to show that if $\bigcap G_n$ is not dense in X then $int(\overline{\bigcap G_n})$ is also not dense in X. I have tried ...
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1 vote
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### Solution set of any polynomial in $\mathbb{R}[x,y]$.

Lee in Introduction to Topological Manifolds claims that the solution set $X(f)$ of any polynomial $f\neq 0$ in two variables over $\mathbb{R}$ is nowhere dense i.e. $A:=\mathbb{R}^2-\overline{X(f)}$ ...
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### Given an open set $U \subset \mathbb{R}$, is it possible to express $U$ as a disjoint union of $(a,b)$'s where $a,b \in Q$ for some dense set $Q$?

I am aware that any open subset $U \in \mathbb{R}$ may be expressed as a countable disjoint union of open intervals. Now, if we are given a dense subset $Q \subset \mathbb{R}$, is it possible to ...
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1 vote
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### Conditions such that complement of level set in smooth manifold is dense

Let $M$ be a smooth (hausdorff, second countable) manifold of dimension $n\geq 2$ and consider a $C^\infty$ function $f:M\to\mathbb R$. Q: Under what conditions is $S = M\setminus f^{-1}(0)$ is dense ...
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### How does the reflexivity of $V$ imply the denseness of $R(i^*)$?

For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. I'm reading about Gelfand triple at page 136 in Brezis' Functional Analysis. Let $(H, \langle \cdot , \cdot \rangle_H)$ ...
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1 vote
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### When can monothetic groups be turned into rings?

If G is a Hausdorff topological group, saying that G is monothetic is equivalent to saying there exists a homomorphism $f: \mathbb{Z} \to G$ with dense image. A multiplication can be naturally defined ...
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### Nowhere Dense Sets in Discrete Metric Space

Not Homework- Just Personal Study Let $(M,d)$ be a discrete metric space. I want to show that the only subset of $M$ that is nowhere dense is the empty set. Let $E$ be a non-empty subset of $M.$ The ...
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### Nowhere Dense Set and Corollary to Baire's Theorem

Not Homework just Personal Study Reference text: Kolmogorov and Fomin - Introductory Real Analysis I am working on understanding why a complete metric space with no isolated points is uncountable. The ...
• 71
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### There are dense orbits in the set of all allowed sequences

Lately, I have done an exercise in the book "Introduction to Dynamical Systems" by Brin and Stuck. Exercise 1.4.5: Assume that all entries of some power of $A$ are positive. Show that in ...
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### $x, Ax, \cdots, A^mx, \cdots$ is not dense in $\Bbb R^n$, where $A$ is a $n\times n$ real matrix, $x\in\Bbb R^n$.

$x, Ax, \cdots, A^mx, \cdots$ is not dense in $\Bbb R^n$, where $A$ is a $n\times n$ real matrix, $x\in\Bbb R^n$ are fixed. This is a problem involving mathematical analysis and higher algebra. I ...
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### Prove that set of matrices is dense in $U(2)$

Consider the group of matrices $B$ generated by taking products of the matrices \rho_1 = \begin{pmatrix}\exp(-4\pi i/5) & 0\\ 0 & \exp(3\pi i/5)\end{pmatrix}\\ \rho_2 = \begin{...
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### Connected subgroups of unit circle

It's well known that any subgroup H of $\mathbb{S}^1$ is either dense or finite. Therefore, if H is compact, it implies that H either is finite or equal to the entire unit circle. My question is ...