All Questions
13,925 questions
6
votes
1
answer
442
views
Countable paracompactness, normality and locally countable open covers
(repost from the topology Q&A board)
I have a (T_1), Normal, countably paracompact space X. I would like to know if every locally countable open cover of X (i.e. an open cover such that every x ...
- 5,407
3
votes
2
answers
467
views
Euler characteristics and operator indices as exponents for Laurent polynomials
This question is rather vague. Are there any natural situations which involve Laurent polynomials of the form
$$\sum q^{a_i}\in\mathbb{Z}[q,q^{-1}]$$
where the $a_i$'s are either Euler characteristics ...
- 3,474
1
vote
1
answer
2k
views
spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]
Let a,b be 2 elements in a Banach Algebra.Let Spec(x) denote the spectrum of an element x. If a,b commute with each other, then by Gelfand Transformation, we have Spec(a+b) is a subset of Spec(a)+Spec(...
- 7,329
6
votes
2
answers
497
views
Can I detect the point of impact without looking at it?
I'm going to postpone the motivation for this question because the question itself involves no complicated maths and may well have a very simple solution so I don't want to put anyone off with high ...
- 23.8k
1
vote
1
answer
994
views
On the convolution of generalized functions
It is provable that $f_\lambda\to f\Rightarrow f_\lambda*g\to f*g$ if $g$ has a compact support (shown in my textbook). In my particular case, $g=u(t+\triangle t)-u(t-\triangle t)$. Does for that ...
- 26.8k
5
votes
7
answers
4k
views
Do the empty set AND the entire set really need to be open? [closed]
My question is motivated by the previous discussion 'Why is a topology made of open sets?'. While the axioms for arbitrary unions and finite intersections are without doubt essential to the concept of ...
- 65.4k
2
votes
0
answers
223
views
Is the realization of a proper map of simplicial spaces proper ?
Let $f:X \rightarrow Y$ be a map of $m$-dimensional simplicial spaces (which means that all simplices above dimension $m$ are degenerate). Recall, that $f$ is a natural transformation of functors from ...
- 11.1k
4
votes
0
answers
296
views
What is enough to conclude that something is a CW complex (part II)?
A while ago I asked a question about recoqnizing CW complexes and got an extremely nice and concrete answer. However, I am still interested in a more general treatment of this and therefore pose the ...
CommunityBot
- 1
-3
votes
2
answers
314
views
Dispensing with the notion of infinity for the sake of coverings [closed]
Instead of taking a one to one correspondence meaning each set has the same number of elements. why not use the concept of coverings of topology? The irrational numbers covers the whole numbers but ...
- 9,201
2
votes
0
answers
270
views
Homotopy equivalences and cores
Hi all,
Before asking my question, I need to fix some terms and notation.
Let $M$, $M'$ be locally compact, Hausdorff spaces, and $f:M\rightarrow M'$ a homotopy equivalence with homotopy inverse $g:...
- 93
3
votes
3
answers
699
views
Is there a name for this property of a topology?
This property seems like it should have a nice name, but I can't find one anywhere. Does anyone know a name for this?
For each non-empty open set $U$, there exist proper open subsets $\{U_i\}_{i\in ...
- 115
3
votes
1
answer
3k
views
Is a proper quotient map closed ?
I am trying to produce closed quotient maps, as they allow a good way of creating saturated open sets (as in this question).
A map $f:X\rightarrow Y$is called proper, iff preimages of compact sets ...
CommunityBot
- 1
0
votes
3
answers
248
views
how slow can the dimension of a product set grow?
Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:
$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,
where $\sim$ denotes "...
- 11.1k
10
votes
2
answers
3k
views
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?
- 4,736
2
votes
2
answers
584
views
A proof about an unconditional basis theorem
Hello everyone. I'm in a little trouble trying to find the proof of a theorem stated by W. T. Gowers. It is the Lemma 1.6 in his article 'An infinite Ramsey theorem and some Banach space dichotomies' (...
- 31.5k
9
votes
1
answer
718
views
What is enough to conclude that something is a CW complex?
This question was something I considered when looking into CW-structures on Grassmannians, but I found no general treatment of this in the literature:
Question: Assume that $X$ is an $n-1$ ...
- 27.2k
12
votes
2
answers
2k
views
Is this a known compactification of the natural numbers?
Given two infinite sets $A$, $B$ of natural numbers, write $A\preceq B$ if $B\setminus A$ is a finite set. Define the equivalence relation $A\sim B$ if $A\preceq B$ and $B\preceq A$, and let $\partial\...
CommunityBot
- 1
7
votes
2
answers
419
views
Relation between $KO$ and $K$
What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. ...
- 15.4k
1
vote
2
answers
686
views
Existence of convergent subsequences for all values in range?
Consider sequence $s(n) = \sin{nx}$. Are there values of $x$ for which the following holds: For every $y \in \[-1,1\]$ there is a subsequence of $s(n)$ converging to $y$? (Or perhaps just for the open ...
- 1,214
2
votes
3
answers
4k
views
Show a linear operator is not compact
For $f\in L^2(0,\infty),$ define $(Tf)(x)=x^{-1}\int_0^x f(s)ds,$ for $x\in(0,\infty),$ then from hardy's inequality, $T\in B(L^2),$ my question is how to show that $T$ is not compact?
- 4,060
8
votes
1
answer
381
views
Estimating flat norm distance from a planar disc
Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
- 45k
2
votes
1
answer
1k
views
Finding saturated open sets
Suppose I have a continuous map $f:X\rightarrow Y$. Then one can wonder, whether for every open set $U\subset X$ the set
$U':=\{x\in X|f^{-1}(f(x))\subset U\}$ is open again. This is not true in ...
- 11.1k
6
votes
2
answers
2k
views
References and applications involving the Krull Toplogy
I was wondering if anyone can suggest a reference which treats the Krull topology. Most of the books I have found don't go into any kind of detail.
It is my understanding that the Krull topology ...
- 50.6k
2
votes
2
answers
317
views
Bibliography for topologies defined by a family of seminorms
Hello
I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource.
Thank you very much.
- 27.7k
10
votes
2
answers
367
views
existence of a connected set with given connected projections.
Suppose A and B are compact connected sets in the XY plane and XZ plane respectively in R^3. Suppose further that the the range of x-values taken by A and B are the same (i.e, projections of A and B ...
- 45k
5
votes
1
answer
1k
views
Orthogonal complements in Hilbert bundles
It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...
- 26.8k
6
votes
1
answer
1k
views
Reference needed for: every idempotent in a C*-algebra is similar to a hermitian one
The result stated in the title is thoroughly standard - or that's the impression I got.
I seem to remember seeing it stated somewhere in a book I was reading in the library, and then reverse-...
- 7,329
0
votes
1
answer
288
views
The Quantum Operations On The Bipartite Systems
Given two distinct and noninteracting quantum mechanical
systems $\mathfrak{S}\_1$ and $\mathfrak{S}\_2$ with state spaces
$\mathcal H\_1$ and $\mathcal H\_2$, respectively, the state space
of the ...
- 6,342
4
votes
1
answer
2k
views
Fiber bundle = principal bundle + fiber?
This question is heavily related to this question.
Fix a sufficiently nice and connected topological space $B$ and let $FB$ be the category of fiber bundles over $B$. A morphism $f: (E\to B)\to (E'\...
CommunityBot
- 1
3
votes
3
answers
2k
views
Error analysis of implicit functions
I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms):
$$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...
- 15.4k
2
votes
8
answers
3k
views
The core question of topology
As I see it, the core question of topology is to figure out whether a homeomorphism exists between two topological spaces.
To answer this question, one defines various properties of a space such as ...
- 2,927
2
votes
2
answers
1k
views
Simple question of topological cofibration
I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the ...
- 8,803
5
votes
1
answer
467
views
Info about Elton–Odell theorem
Hello everyone, could anyone please tell me where can I find information about the Elton–Odell theorem?
It states:
For any infinite dimensional Banach space $X$ there is a $q > 1$ so that $X$ ...
- 9,366
31
votes
7
answers
5k
views
Why is it useful to classify the vector bundles of a space?
It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces. Now, I've read somewhere that it is ...
- 29.1k
7
votes
2
answers
2k
views
Intersection form in twisted homology (homology with local coefficients)
The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...
- 57.6k
11
votes
5
answers
1k
views
Confusion over a point in basic category theory
"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological ...
- 65.4k
4
votes
1
answer
822
views
What is the tensor product of $L^p(\bf R)$ with $L^q(\bf R)$?
I'm wondering: What is the tensor product of $L^p({\bf R})$ with $L^q({\bf R})$?
(For p=q=2, the answer clearly should be $L^2({\bf R}^2)$; for other values of $p$ and $q$, it is not at all obvious ...
- 11.4k
4
votes
1
answer
417
views
"Category" of Nonempty Metric Spaces and Contractive Maps?
The usual way of getting a category of metric spaces is to take metric spaces as objects, and the nonexpansive maps (ie, functions $f : A \to B$ such that $d_B(f(a), f(a')) \leq d_A(a, a')$) as ...
- 26.8k
3
votes
3
answers
584
views
Polynomials and L^p(R)
As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are no non-trivial ...
- 2,283
3
votes
1
answer
1k
views
characterization of continuous functionals in weak-star topology
Reading Wojtaszczyk's Banach spaces for analysts, I'm trying to understand his proof that the space of all continuous linear functionals on $(X^\star,\sigma(X^\star, X))$ is $X$.
To show the $ \...
- 31.5k
4
votes
2
answers
340
views
Embeddings of Weighted Banach Spaces
Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces
$$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^...
- 25.8k
5
votes
1
answer
403
views
Nonlinear Nuclear Operators ?
Is there a "right" definition of the nuclear
operator in the nonlinear framework ? Of course, such an operator
must be compact, while a linear operator should be "nonlinearly"
nuclear iff it is ...
- 31.5k
24
votes
1
answer
2k
views
How many ways are there to globalize Harish Chandra modules?
Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie ...
CommunityBot
- 1
-3
votes
2
answers
1k
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Finite versus infinite on non-Hausdorff topologies [closed]
Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a ...
CommunityBot
- 1
2
votes
1
answer
336
views
Topologies making a class of functions continuous [closed]
Let $X:=\{f: \mathbb{C}\to \mathbb{C}\}$ be a class of total functions on $\mathbb{C}$ closed under composition, addition, multiplication, and scalar multiplication. Does there exist a topology on $\...
- 236k
8
votes
3
answers
606
views
Compact Hausdorff and C^*-algebra "objects" in a category.
This is yet more on "algebraic objects in functional analysis".
Since Compact Hausdorff spaces are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable ...
- 5,827
16
votes
0
answers
1k
views
Finite Rank Commutators
My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...
- 31.5k
5
votes
3
answers
1k
views
Functional calculus for direct integrals
Suppose I have a direct integral of Hilbert spaces $H = \int^\oplus H_x dx $, and suppose I have an operator $T: H \to H$ which is decomposable, and so it can be written as
$T = \int^\oplus T_x$ for ...
- 1,540
3
votes
3
answers
728
views
What do you call the product of a circle and an annulus?
What would you call the product of an annulus and $S^1$ (a 'thickened' torus like 3-manifold)?
More generally, is there an archive or list online of names assigned to various (non-standard) manifolds ...
- 345
7
votes
3
answers
1k
views
The continuous as the limit of the discrete
Reading this documment: www.math.ucla.edu/~tao/preprints/compactness.pdf, I got interested in the following thing: "One can also use compactifications to view the continuous as the limit of the ...
- 57.6k