gn.general topology – Topological properties inherited by the Hausdorff metric space

Let $(X,d)$ be a metric space and $(K_X , h_d)$ be the associated metric space of compact subsets of $X$ with the Hausdorff metric. It is well known that $K_X$ inherits certain topological (and analytic) properties from $X$. For example, if $X$ is compact, then so is $K_X$; and if $X$ is complete, then so is $K_X$.

Is there a reference which further explores the properties that $K_X$ inherits from $X$? In particular, if $X$ is locally compact then is $K_X$ also?

ag.algebraic geometry – Topological information in sheaf cohomology?

Let $X$ be a projective complex manifold and $Eto X$ a holomorphic vector bundle. When $E=Xtimesmathbb{C}$ there is an isomorphism $$H^n(X, mathcal{O}(E))to H^n(X,mathbb{C})$$
for any $n$. This isomorphism can even be made canonical (in a certain sense).

The question is about possible maps
$$H^n(X, mathcal{O}(E))to text{some “topological” group}$$
in cases when $E$ is not trivial. (We assume that $X$ is simply connected.)
It seems unlikely that topological invariants of this kind exist in general,
but I am curious if there is anything interesting in special cases.

fa.functional analysis – Strong topology on a topological vector space

I’m not sure this is an appropriate question for this site but I’ve tried math stack exchange and got no answers. Also, this problem arose in one of my research problems, so I’m stating it here.

The strong operator topology is defined on Simon’s book as follows. It is the weakest topology on $mathcal{L}(X,Y)$ such all the maps $E_{x}: mathcal{L}(X,Y) to Y$ defined by:
$$E_{x}(T) := Tx $$
are continuous for all $x in X$. Here, $X$ and $Y$ are supposed to be Banach spaces and $mathcal{L}(X,Y)$ is the space of all bounded linear operators from $X$ to $Y$. A neighboorhood basis for this topology, in Simon’s words, is given by the sets of the form:
$$ {S: hspace{0.1cm} Sin mathcal{L}(X,Y), hspace{0.1cm} ||Sx_{i}||_{Y}<epsilon, hspace{0.2cm} i=1,…n}$$
where $x_{1},…,x_{n}$ is any finite collection of elements of $X$ and $epsilon > 0$.

I know the notion of strong topology can be extended to more general spaces such as topological vector spaces, but I don’t want to get too deep into the theory. However, I’m interested in the case where $X$ is not Banach but $Y = mathbb{C}$ is Banach.

My question is: In my setup, if $X$ is a Fr├ęchet space and $Y=mathbb{C}$ is Banach, the above definition seems to work just fine if I replace $mathcal{L}(X,Y)$ the space of bounded linear operators to its analogue, the space of all continuous linear maps. The same properties seem to hold in this case. Is it a correct definition of a strong topology to my particular case? In other words, if I was to consider $X$ as a topological vector space and $X^{*}$ its topological dual, would the strong topology defined on $X$ be the same topology I’m proposing?

Does the compact-open topology retain topological groups?

Let $X$ be a topological space and $Y$ a topological group. Then $C(X,Y)$ is a group, and can also be endowed with the compact-open topology.

Is $C(X,Y)$ in the compact-open topology necessarily a topological group? If not, is there some property of $X$ which will guarantee it?

general topology – Open subset of Topological Space

If I have some topological space $X$ with topology $mathcal{T}$, is any open subset of $X$ a topological space with the same topology.

I’m just starting to teach myself about topological spaces. After looking at the open set and neighborhood definition I was wondering if this is true. I feel like it is but I’m not sure how to justify that. Any help is much appreciated!

Topological spaces Open and closed sets

Let A and B – respectively closed and open sets in topological space T. Proof that A B is closed and B A is open.

topological dynamics – A reference to the fact that a topologically transitive action of a group on a compact metric space has a dense orbit

I need an appropriate reference to the following obvious fact:

A group action $ G $ on a compact metrable space $ K $ is topologically transitive (= the orbit of any non-empty open set is dense) if and only if it is the orbit $ Gx $ from a certain point $ x in K $ is dense in $ K $.

For a cyclic group, this characterization is proven here. I hope that some textbooks on topological dynamics should contain such a basic fact. Help me!

sequences and series – Clarification on the topological proof of Ferstenburg "Infinitude of Primes"

I am fairly new to topology and am particularly interested in gaining an intuitive understanding for the following proof:

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I wonder if anyone could slow down the sequence of thoughts here so that I can put the puzzle together more. For example:

  1. In what sense is the topology described "metriable"

  2. How arithmetic progressions $ (- infty, infty) $ to be both open and closed (and I don't really understand why this is implied via the complement of the union). Therefore, why does this imply the closure of finite progressions?

  3. How does all of this help build the picture of the final conclusion.

I "agree" with the basics of topology / measurement theory / diff. geo – just in case you need to assess how much you need to tailor the answer.

Theory of gr.groups – Does a cocompact subgroup of a topological grouo contain a normal cocompact subgroup?

Motivation: It is obvious that for a subgroup with finite index $ H $ of a group $ G $, there is a normal subgroup $ K $ of $ G $ $ K subset G $ and $ | G / K | < infty $

Our question: Let $ G $ to be a topological group and $ H $ be a closed normal subgroup but not necessarily $ G $ so that the topological space quotient $ G / H $ is a compact space. Is there a closed subgroup $ K subset H $ which is normal $ G $ and $ G / K $ is a compact topological group.

Topological sorting and related to dag

How to construct an acyclic graph directed from its p no of given topological types if the external degree of each is at most equal to 2?