## General topology – Bijection between continuous maps \$ B to S \$ and open subsets of \$ B \$

Is my solution of the following problem correct?

Let $$S$$ the two-point topological space in which only one of the singleton subsets is open. Prove that continuous maps from a space $$B$$ at $$S$$ correspond bijectively to open subsets of $$B$$.

Let $$S = { star, bullet }$$ and let $${ bullet }$$ to be the singleton subset that is open.

Set the map $$F$$ of maps $$B to S$$ open subsets of $$B$$ by sending a card cts $$phi: B to S$$ to the subset $$phi ^ {- 1} ( { bullet }) ​​$$. This is an open subset of $$B$$ because $$phi$$ is continuous.

Define the map in the opposite direction by assigning to an open subset $$U$$ of $$B$$ function $$psi_U: B to S$$ who sends each element of $$U$$ at $$bullet$$ and each element outside of $$U$$ at $$star$$. This map is continuous because the pre-image of each open subset is open: the pre-images of $$emptyset$$ and $$S$$ are, respectively $$emptyset$$ and $$B$$ (which are open), and the preimage of $${ bullet }$$ is the open subset $$U$$.

Let us check that $$FG = 1$$ and $$GF = 1$$.

1) $$F (G (U)) = F ( psi_U) = psi_U ^ {- 1} ( { bullet }) ​​= U$$.

2) $$G (F ( phi: B to S)) = G ( phi ^ {- 1} ( { bullet })) = psi _ { phi ^ {- 1} ( { bullet })}$$. To see that $$phi = psi _ { phi ^ {- 1} ( { bullet })}$$, Note that $$psi _ { phi ^ {- 1} ( { bullet })}$$ send to $$bullet$$ these and only these elements $$x$$ For who $$x in phi ^ {- 1} ( { bullet }) ​​$$ (that is to say, for which $$phi (x) in { bullet }$$, that is, for which $$phi (x) = bullet$$) – That's it, $$psi _ { phi ^ {- 1} ( { bullet })}$$ send to $$bullet$$ precisely the same elements that $$phi$$ send to $$bullet$$, and $$psi _ { phi ^ {- 1} ( { bullet })}$$ sends the other elements to $$star$$ (which is valid for $$phi$$ as well as). So $$phi = psi _ { phi ^ {- 1} ( { bullet })}$$.

## Gt.geometric topology – Limitations on triangulation

Let $$E$$ to be the total space of a linear $$k$$-disk bundle on a closed smooth collector $$B$$.

Yes $$B$$ has a smooth triangulation with $$m$$ simplices, can $$E$$ be triangulated smoothly with $$N$$ simplices, where $$N$$ depends only sure $$k$$ and $$m$$?

Note: It seems that I can prove the assertion weaker than $$k$$, $$m$$
there are only a lot of possibilities for the type of homeomorphism $$E$$ but the proof goes well beyond the PL topology, which I find confusing.

More generally, we could ask the same question when $$E$$ is a regular neighborhood of a sub-complex $$B$$ but in this case, I do not know how to prove the topological finality for $$E$$.

## Geometric topology – Wild character of the codimension 1 subvarieties of the Euclidean space

This question arose from this pile trading post. I am writing a thesis on the $$s$$-Cobordism and Siebenmann's work on the end obstructions. Combined, they give a quick proof of the uniqueness of the smooth structure for $$mathbb {R} ^ n$$, $$n geq 6$$. Siebenmann's theorem says pretty much that for $$n geq 6$$ a contractible $$n$$-collecteur $$M$$ which is simply connected to infinity integrates (smoothly) inside a compact collector. Since this compact variety is contractible, by the $$s$$-cobordism, it is diffeomorphic to the standard $$n$$-disk $$D ^ n$$ (see Minor Conferences on the $$h$$-cobordisme for example). It follows that $$M = text {int} D ^ n$$ is diffeomorphic to $$mathbb {R} ^ n$$.

The problem is that the case $$n = 5$$ is not covered. I am aware of Stallings beautifully written On the piecewise linear structure of the Euclidean space but I'm looking for a way to deal with the $$n = 5$$ case via Siebenmann's end theorem and the good $$s$$theorem of -cobordism (see link to the question mse). This brings me to the next question, which is interesting in itself

Since codimension 1 is well integrated and smooth $$S subset mathbb {R} ^ {n + 1}$$, is there a diffeomorphism auto $$mathbb {R} ^ {n + 1} rightarrow mathbb {R} ^ {n + 1}$$ who wears $$S$$ in a region limited to one dimension $$mathbb {R} ^ n times (-1, 1)$$ ?

Now if $$M$$ is a multiple that is homeomorphic to $$mathbb {R} ^ 5$$, the product $$M times mathbb {R}$$ is homeomorphic to $$mathbb {R} ^ 6$$, and therefore also diffeomorphic. Subject to the existence of diffeomorphism in my question, we could find a diffeomorphism $$f: M times mathbb {R} rightarrow mathbb {R} ^ 6$$ that cards $$M times 0$$ in $$mathbb {R} ^ 5 times (-1, 1)$$. This would produce a good $$h$$-cobordism between $$M$$ and $$mathbb {R} ^ 5$$ taking the area between $$f (M times 0)$$ and $$mathbb {R} ^ 5 times 1$$ in $$mathbb {R} ^ 5 times mathbb {R}$$. Since $$M$$ is simply connected, the good $$s$$-cobordism theorem applies and shows that $$M$$ and $$mathbb {R} ^ 5$$ are really diffeomorphic.

## Algebra Topology – Abandoning Mathematical and Suicidal Thoughts

Dear wise mathematicians,

I am an undergraduate student in mathematics at UW-Madison. I do not know if this message is acceptable here, but I want to mention a recent event that almost forced me to commit suicide.

I am an aspiring topologist who is very interested in algebraic topology, geometric topology, theoretical topology of sets and their potential links with the study of prime numbers. My future goal is to really connect topology and analytic number theory, which I think has not been rigorously studied, contrary to the algebraic theory of numbers. With two faculty mentors (topologist and analyst), I studied the potential use of topology to study Goldbach's conjecture; I've been able to create a topological space that captures the GC in finite terms and create a ZFC + model designed to attack the GC in a specific way. My mentors really love this idea and we have been chasing them ever since.

There is another topologist on campus (he retired a long time ago) that I respect and really look at. He is well known for his work on differential topology and geometry (one of the founders of differential topology). I often discuss mathematics with him and I even introduced him to my ideas about GC's attack. I recently learned that he had mentioned to friends that I was completely unfit to do math and that I had to stop doing it. He also criticizes my ideas as being stupid and I may have stolen them from other teachers (which I did not do myself for several months to develop before introducing them to my mentors). , and I know that there is no article published about them). When I learned this, I was totally destroyed … He is incredibly famous in mathematical communities and he is very old, armored with wisdom and intuition. Since wise mathematicians like him think I am completely incapable of doing mathematics, should I listen to his thinking and just stop studying mathematics? I am very shocked to the point of losing all confidence in myself and I see no desire to continue my life, so suicide is the solution … I do not know what to do!

My friends told me to give everything I had on my research, to solve GC and prove to him that he was wrong, but they are not involved in mathematics.

A side note, I have been suffering from depression for a long time (and I get professional care to treat it). I know that depression has had an impact on my courses and my studies, but studying mathematics (especially topology) is the only way for me to escape the influence of lows on me. Given that I am suffering from severe depression, is not it a good idea to pursue mathematics studies in graduate programs and to do research in the future?

## General Topology – Topological Conjugation of a Solenoid in a Solid Toroid

Let $$S ^ 1$$ to be the unitary circle and $$B ^ 2$$ be the unit disk (closed) in the plane. The Cartesian product $$D = S ^ 1 times B ^ 2$$ is a strong torus in$$R ^ 3$$. Consider the map $$F: D to D$$, $$F ( theta, p) = (2 theta, frac {1} {10} p) + frac {1} {2} e ^ {2 pi i theta}$$. Let $$Lambda = bigcap limits_ {n geq 0} F ^ n (D)$$, and $$Sigma = { theta = ( theta_0 theta_1 theta_2 …) | theta_j in S ^ 1 and g ( theta_ {j + 1}) = theta_j) }$$,or $$g: S ^ 1 to S ^ 1, g ( theta) = 2 theta$$.
Set a metric on $$Sigma$$.Yes$$Theta = ( theta_0 theta_1 theta_2 …)$$ and $$Psi = ( psi_0 psi_1 psi_2 …)$$, we define the distance between them to be $$d ( Theta, Psi) = sum limits_ {j geq0} frac {| e ^ {2 pi i theta_j} -e ^ {2 pi i psi_j} |} {2 ^ j}$$
Set an offset $$sigma: Lambda to Lambda, sigma ( theta_0 theta_1 theta_2 …) = (g ( theta_0) theta_0 theta_1 theta_2 …)$$then $$sigma$$ is a homeomorphism.

Now let $$pi: D to S ^ 1$$ to be the natural projection, that is to say $$pi ( theta, rho) = theta$$. For any point $$x in Lambda$$, the map $$S: Lambda to Sigma$$ given by$$S (x) = ( pi (x), pi F ^ {- 1} (x), pi F ^ {- 2} (x), …)$$is well defined and $$S circ F = sigma circ S.$$

How to show that: $$S$$ gives a topological conjugation between $$F$$ sure $$Lambda$$ and $$sigma$$ sure $$Sigma$$ ?

This is from Robert L. Devaney's book $$An introduction to Chaotic Dynamical systems$$ second edition of forms P201 to P208.

## Why does longitude correspond to Frobenius in arithmetic topology and other strange phenomena?

I'm trying to address Morishita's book Nodes and Premiums to learn a little more about arithmetic topology, but some analogies surprise me. I know that correspondence has to be addressed with a grain of salt, but some parts are so fundamental that I would like to understand them better.

1. In the table (3.3) on page 50 of his book, Morishita writes that longitude, called $$beta$$, should correspond to a Frobenius lift and meridian, called $$alpha$$, corresponds to a taming inertia generator (both as elements of the maximum taming quotient of the absolute Galois group of a local field). It calls "longitude" a path that bypasses a hole in the boundary of a tubular neighborhood of the node and "meridian" the edge of a disc that is a "cross section" of the tube. If the knot was the denouement, this neighborhood would be the complete torus $$S ^ 1 times D ^ 2$$: in that case, $$alpha = partial D ^ 2$$ and $$beta = S ^ 1$$. This analogy does not support my view that inertia acts as a monodromy, which is "turning around holes", but I tried to pursue it. Then (page 63, after Theorem 5.1), he describes the analog of decomposition groups for an uninterpreted node. $$K$$: he says that this group should be generated by "a circulating loop" $$K$$"which I think is just the image of $$alpha$$. Then I'm completely lost, because I would expect Frobenius to generate decomposition groups in un-simplified situations …
2. In chapter 11, an experimental analogy with the Iwasawa theory is suggested. Nevertheless, it seems to me that something is odd, because the typical Iwasawa theory concerns a very wild branch, whereas the same table (3.3) on page 50 seems to indicate that there is no wild topological inertia. So the Galois group of $$X_ infty / X_K$$, or $$X_K$$ is a node complement and $$X_ infty$$ is a $$mathbb {Z}$$-covering $$X_K$$, in a sense, resembles an "infinitely thinly branched cover" (which has no arithmetic analog) rather than a $$mathbb {Z} _p$$-extension. Am I missing something? In the same sense, he has a small parenthesis between p. 144 and p. 145 where he writes that "[Ensupposantqu&#39;a[assumingthata[ensupposantqu'un[assumingthata$$mathbb {Z} _p$$-extension be branched to a first only, and this first be totally ramified]is a hypothesis analogous to the case of knot. "Why is it so? Or the fact that one $$mathbb {Z}$$-cover must be branched to a single node, and the fact that there can not be a small, unbranched layer below seems obvious to me (at least if the basic variety is not $$S ^ 3$$, other $$pi_1 (S ^ 3) = 0$$ should say there is no unassembled extension).

## Algebras oa.operator – Equivalence of the \$ sigma-weak topology with respect to another topology

Let $$mathcal H$$ to be a Hilbert space. Define a topology $$tau_1$$ sure $$B ( mathcal H)$$ by the seminorms family $$x mapsto | Tr (xa) |,$$ $$a in L ^ 1 (B ( mathcal H)).$$ Right here $$B ( mathcal H)$$ refers to the set of all linear maps delineated on $$mathcal H$$ and $$L ^ 1 (B ( mathcal H))$$ refers to the operators of the trace class.
Again define the $$sigma$$Topology -WOT $$tau_2$$ sure $$B ( mathcal H)$$ by removing the weak operator topology from $$B ( mathcal H otimes ell_2)$$ at $$B ( mathcal H)$$ via the map $$x mapsto x otimes 1.$$ How to show that $$tau_1 = tau_2$$? In many books and lecture notes in von Neumann's algebras, they have just mentioned that this is true. But I could not find any solid proof.

## general topology – A question on a detail of Eberlein-Smulian's theorem: the relationship between weak and strong closures of a set

Theorem:

Let $$B$$ to be the closed unit ball of a Banach space $$X$$. then $$B$$ is weakly compact if and only if it is weakly compact sequentially.

Forward:

We suppose first of all $$B$$ is compact. Kakutani's theorem tells us that $$X$$ is reflexive.
Each sequence delimited in $$X$$ has a weakly convergent subsequence. Since $$B$$ is weakly closed, $$B$$ is weakly compact in sequence.

Why can we assume $$B$$ is weakly closed? I guess that stems from $$B$$ to be strongly closed, but I'm not clear about the relationship of the two.

I think that a weak closure implies a strong closure: $$B$$ is weakly closed and his compliment is therefore weakly open. So $$B ^ C$$ is certainly open in a stronger topology – such as the standard topology. So $$B$$ is closed in the topology of the norm. I do not see how the other direction can be true because we "come from a finer topology".

## General Topology – Frechet Distance Properties

There is an old construction, apparently due to the PhD thesis of Frechet (unfortunately written in French and in ancient notation), which transforms the set of curves of a modulo reparamétrisation in a metric space into a space metric itself. The idea is that if $$f$$ and $$g$$ are curves in a metric space $$X$$ (meaning continuous maps of $$[0,1]$$ at $$X$$), their distance is defined as
$$d (f, g) equiv mathrm {Inf} _ { phi, psi} mathrm {Sup} _ {t in [0,1]} d (f ( phi (t)), g ( phi (t)),$$
or $$phi$$ and $$psi$$ homeomorphisms preserving the orientation of $$[0,1]$$ to himself.
Most properties of a distance derive trivially from the definition, but I find it hard to show that $$d (f, g) = 0$$ implies that $$f$$ and $$g$$ differ only by a homeomorphism. Potentially, we could just have a sequence $$( phi_n, psi_n)$$ homeomorphisms for which the supremum converged to zero, but which did not converge to a pair of homeomorphisms $$( phi, psi)$$ for which he disappears. Why should the infimum be realized? I think it should be possible to show that by using some kind of uniform convergence and possibly the compactness of $$[0,1]$$but have not succeeded. For my own purposes, I am particularly interested in whether or not to use the compactness of $$[0,1]$$ (I'm trying to understand how far $$[0,1]$$ can be generalized to an arbitrary topological space in this definition).

## hereditary topology to subspaces

assume $$X$$ is an arbitrary subset of a topological space $$Y$$ with a strong topology, the topology induced on $$X$$ is also strong topology.