## Topological spaces.

If X is a set. that T and V are two topologies on X and V are finer than T. Someone can make me understand this with the help of a diagram, I am novice in this field. Thank you .

## Re-label in the foreground, what does the topological type mean for an eligible network?

In the CLRS book, it is indicated that the algorithm maintains a list of vertices sorted topologically in the allowable network and that the vertices with excess zero flow are moved to the beginning of the list.

I do not quite understand what that means. I imagine that the vertices are sorted according to the number of incidentable edges incidental. But then, how does moving vertices without excess flow towards the front affect the sort order in this case? Moreover, how is it that the list is already sorted when it is initialized with a random order of vertices?

## Mathematical Physics – What do physicists mean by a topological theory of quantum gravity?

This is a matter of jargon.

The fact that this is posted here rather than in a physics forum indicates two things

1. I know too little physics.
2. An explanation with more mathematical flavors will be appreciated more ..

### Context

I should first explain what is a theory of gravity in my imagination: it seems to be a theory that governs the relationship between space and matter. For example, Hilbert has solved this problem by introducing a function on the metric space, which results in Einstein's field equations which concern the curvature of space-time and the mass-energy-moment tensor.

A quantum field theory seems to me (in my eyes) to be a theory of fields where each field could possibly occur. In our case, the fields are the metrics, whose amplitudes can be calculated by a "quantified" action weight

### Question

This brings us to the confusing part: a topological theory (seems) to mean a theory that does not depend on geometry (in particular, metrics)! What does a topological theory of quantum gravity mean?

## Is there a class of topological spaces such that any arbitrary intersection of closed sets is not empty?

Is there a class of topological spaces such that any arbitrary intersection of closed sets is not empty?

## general topology – Assumption of Absorbent Sets in the Topological Vector Theorem

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.

## 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.