My question concerns this continuity of multiplication in a topological vector space. In concrete terms, it seems to me that the answer continues to be correct if we consider $ W = N times (-1,1) $. However, in this case, I do not think the games are absorbing, because step 3 of the answer is a consequence of the fact that the games are balanced.

# Tag: Topological

## ra.rings and algebras – Topological Brauer group and sheaf of $ C ^ { infty} $ – functions

Given a smooth variety $ X $; by definition, the topological group of Brauer $ B (X) $ is the class group of Azumaya algebras on the sheaf $ mathcal {O} _X $ of $ mathbb {C} $functions evaluated on $ X $.

If we replace the sheaf $ mathcal {O} _X $ by the sheaf of $ C ^ { infty} $-functions; Will the resulting Brauer group be the same?

## Ag.algebraic geometry – $ G $ -torsor for the topological space compared to that of the cluster of groups

I just read about the definitions on the wrestling torsor of groups and get a little confused.

How does the notion of $ G $-torsor for a topological space compared to that of a sheaf of groups? Is there a similar weak equivalence $ Omega B G simeq $ G for $ G $ a group shower?

Why is there equivalence? $ Omega BTors (G) simeq G $?

## sort – How to make an inverted topological sorting using depth search first?

I replace the venerable `make`

utility that will support, among other things, automatic cleaning. The utility automatically determines which files and directories are targets, and then deletes them if the user wants to perform a cleanup operation. However, a file can reside in an automatically created directory, which means that I should perform topological sorting of targets. Each file has an arc to the parent directory and each directory has an arc to its parent. So, for example:

`objhierarchy/obj/foo.o`

has a bow towards`objhierarchy/obj`

`objhierarchy/obj`

has a bow towards`objhierarchy`

What complicates things is that the files must be deleted in the reverse order. So, in the given example, you need this order: (1) `objhierarchy/obj/foo.o`

, (2) `objhierarchy/obj`

, (3) `objhierarchy`

.

The topological type seems to be a good solution, but it gives the opposite order. Thus, a topological kind of the parent graph of the repertoire would give (1) `objhierarchy`

, (2) `objhierarchy/obj`

, (3) `objhierarchy/obj/foo.o`

.

One solution might be a reverse pointer buffer in place (or simply iterated in the reverse order), but I would like to avoid allocating additional memory.

What is the best way to delete files in the reverse order? Can the topological sort algorithm based on the first depth search be modified to call a callback function in reverse order?

## topological groups – How can I prove the following isomorphism?

One of my teachers said the following: $ mathbb {A} _ { mathbb {Q}} $ to be the adele group of $ mathbb {Q} $. There is an isomorphism of topological groups $$ frac { mathbb {A} _ { mathbb {Q}} { mathbb {Q}} simeq varprojlim ( frac { mathbb {R}} {n mathbb {Z}} $ $, where the limit of the right is considered with $ mathbb {N} $ ordered by divisibility. I've tried to make a proof by using both spaces as compact and trying to build a continuous function. However, I had no success. Do you know how to prove it or do you have a clue?

## Discrete and countable topological spaces

I would like to show that a countable discrete space X is locally Euclidean of dimension 0. I know that the way to do this is to show that each point of X has a homogeneous neighborhood at $ mathbb {R} ^ {0} $. The problem is that I do not really know what $ mathbb {R} ^ {0} $ means. I was able to show that each countable separate space has a countable base and is trivially Hausdorff. Therefore, showing that it is locally Euclidean of dimension 0 means that it is a 0-dimensional variety.

## partial order – Does each partially ordered relationship and its dual have the same number of topological ordinances?

Given the Hasse diagram of a partially ordered relationship, is it true that the POSET itself and its dual POSET have the same number of topological ordinances? I have tried some examples and, although that seems to be the case, I need a formal way to prove it. So help me with that.

## 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.

## Topological Graph Theory – Layer Thickness

A book incorporating a G-chart consists of placing the vertices of G on a back and assigning graphic edges to the pages so that the edges of the same page do not intersect. The page number is a measure of the quality of an embedded book that corresponds to the minimum number of pages in which chart G can be incorporated.

If a graph $ G $ is a finite overlay graph $ B $,

is there a relationship between their page number?

I think the cover chart is more complicated than the basic chart. Similarly $ pn (G) geq pn (B) $ hold in general?

## General topology – A definition of differentiable functions for arbitrary topological spaces

**Context**

It is well known that there is no notion of a derivative for **arbitrary** topological spaces. However, by examining the notion of derivative as we find it in a variable, the actual analysis, I managed to generalize the notion. I now wonder what properties this notion of "differentiable function" must satisfy to become a "good" definition.

Our motivation for the formulation of this definition is mainly the definition of Caratheodory derivative.

**The definition**

Before entering directly into the definition itself, consider a particular case of the definition in the following:

Definition 1.Let $ R $ ring and $ tau_1, tau_2 $ to be two topologies on $ R $. A continuous function $ f: (R, tau_1) to (R, tau_2) $ will then be said to be $ ( tau_1, tau_2) $-differentiable on $ R $ at $ a in R $ if there is a function $ g: (R, tau_1) to (R, tau_2) $ such as,$$ f (x) -f (a) = g (x) (x-a) $$for everyone $ x in R $ and $ g $ is continuous to $ a $.

We can generalize the definition above as follows,

Definition 2.Let $ X, Y $ to be arbitrary topological spaces. A continuous function $ f: X to Y $ is said to bedifferentiable with respect to a function $ g: (f (X)) ^ 2 times X ^ 2 to Y $ at $ a in X $if for any net $ (x_ alpha) _ { alpha in J} $ converging towards $ a $, $ g $ is continuous to $ ((f (a), f (a)), (a, a)) $.

For example, if $ X = Y = mathbb {R} $ (equipped with the usual topology) and $ f $ is differentiable to $ a in mathbb {R} $ then we can define, $ g: (f (X)) ^ 2 times X ^ 2 to Y $ as following,

$$ g Bigl ((f (x), f (a)), (x, a) Bigr) = begin {case} dfrac {f (x) -f (a)} {xa} & text {if} ~ x does not \ f (&) text {else} end {cases} $$

Also if $ X = U $ (open in $ mathbb {R} ^ m $) $ Y = mathbb {R} ^ n $ and $ f $ is differentiable to $ to U $ then we can define, $ g: (f (X)) ^ 2 times X ^ 2 to Y $ as following,

$$ g Bigl ((f (x), f (a)), (x, a) Bigr) = begin {case} dfrac {f (x) -f (a) -f & # 39; (xa)} { lVert xa rVert} & text {if} ~ x ne \ 0 & text {else} end {cases} $$

**Question**

What properties should this notion of differentiable function have to be a "good" definition of differentiable functions?