# Tagged Questions

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### Tubular neighbourhood which is nowhere piecewise linear

I recently asked this question. I think, if the following were true, then I would solve my problem. Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...
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### Relationship between weak Lp and strong Lq topologies for q<p

Specificaly: Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence? Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If ...
431 views

### Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point. Is the notion of FPP for topological spaces an absolute notion? More ...
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### Is the multiplicity of a Sobolev mapping alwasys locally essentially bounded？

I have the following question: Let $f:\Omega\to \mathbb{R}^n$ be a (sense-preserving) continuous Sobolev mapping in $W^{1,n}$, where $\Omega$ is a domain in $\mathbb{R}^n$. By Sard's theorem for ...
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### Is the countably infinite product of locally convex topological vector spaces locally convex?

Let $(X,\tau)$ be a locally convex topological vector space and denote the product space $$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$ If we endow $X^{\infty}$ ...
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### Is it always possible to “encircle” exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?

Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points and a positive integer $n$, is there ...
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### Hausdorff metric on C[0,1]

Let us consider $C[0,1]$, the space of continuous functions $f\colon [0,1] \to \mathbb{R}$. It comes usually with the metric of the maximum, or of the supremum, $d_{L^{\infty}}$. Each element $f$ in ...
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### Analytic extension from a closed analytic subset

I have the following question: Let $\Omega \subset \mathbb{R}^{n}$ be an open set and consider $X \subset \Omega$ an analytic subset. By this I mean that there exists analytic functions ...
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### non-trivial convergent sequence [duplicate]

I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$) can you give me a example of ...
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### Special finite subcover of a compact

Let $(a,b)\in \mathbb R^n$. We consider the following open cover of the compact line segment $[a,b]$: $$[a,b]\subset\underset{x\in [a,b]}{\bigcup}B(x,\rho_x),$$ where for $x\in K,B(x,\rho_x)$ is a ...
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### Extending coverings over dense subsets

Let $X$ be a metric space with $D⊆X$ a dense subset. If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$? For a ...
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### The pth power of a distance function is twice continuously differentiable, for $p>2$?

Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$. Is $\beta^p$, $p>2$ a twice continuously ...
786 views

### Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...
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### Baire sets of $X$ possess the required Cartesian product property

Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\mid E_{i}\text{ is a Borel set in }X_{i}\;,\text{ for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the ...
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### Some Questions about zero-dimensional subsets of the unit interval related to cantor set

Let $\mathbb{P}$ denote the set of all irrational numbers in the open segment$(0 , 1)$. let $K$ be the intersection of $\mathbb{P}$ and the standard cantor set and $H=\mathbb{P}-K$. as you know these ...
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### Stable subsets with respect to pointwise convergence.

Consider the linear spacet $\mathcal{F}(\mathbb{R}^n)$ of all real functions defined in $\mathbb{R}^n$. It is well known that the subspace $\mathcal{C}(\mathbb{R}^n)$ of all real valued continuous ...
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### Approximating compact sets with finite sets

Let $(X,d)$ be a metric space. Let $F(X)$ denote the collection of finite subsets of $X$ and let $K(X)$ be the collection of all non-empty compact subsets of $X$. I want to show that $K(X)$ is equal ...
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### On the uncountability of zero sets

If $f$ is any real-valued function, we define its zero set $Z_f = \{ x : f(x) = 0 \}$. Obviously, the zero set of a nice function can be uncountable. e.g., if $f(x) = 0$ on an uncountable domain. I ...
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### Patching together homeomorphisms: how badly can it fail?

Suppose we have a set $X$ with $X=U \cup V$. If we pick a permutation $f$ of $U$ and a permutation $g$ of $V$ which agree on the intersection $U \cap V$, we can coalesce them into one big endo-map $F$ ...
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### Special functions on the unit disk

Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk. We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following: 1) it is a ...
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### Every real function has a dense set on which its restriction is continuous

The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous. Or so I'm told, but this leaves me ...
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### Does Arzelà-Ascoli require choice?

Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version: ...
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### Proofs of Baire category theorem

I would like to have a list of proofs of the fact that the real line is not meager (also very useful would be a reference to such a list, if it already exists somewhere). My motivation is the ...
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### Can there be two continuous real-valued functions such that at least one has rational values for all x?

Of course, no continuous real valued non-constant function can attain only rational or irrational values, but can there be a pair of nowhere-constant continuous functions f and g such that for all x, ...
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### A Jordan Arc in the unit disk

Let D be the open unit disk, and J a Jordan arc (that is a homeomorph of [0, 1]) that lies in D, except J(0) lies on the boundary of D, say J(0)=1. I would like to see that D\J([0, 1]) is a path ...
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### What is the pure intuition for topological continuity and topology? [closed]

I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity. The ...
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### continuous selection of a multivalued function?

The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...
### Continuous function from $[0,1]$ to $[0,1]$
Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?