General Topology – Topological Groups Containing the Sorgenfrey Line

the Sorgenfrey line $ mathbb S $ is the actual line with the topology generated by the base composed of all half-intervals $[Ab)$[Ab)$[ab)$[ab)$ for real numbers $ a <b $.

The Sorgenfrey line is a first accounting and non-metalisable name and is therefore not homeomorphic to a topological group.

On the other hand, the Sorgenfrey line $ mathbb S $ is homeomorphic to a subset of a topological group. For example, the free topological group $ F ( mathbb S) $ more than $ mathbb S $ contains a closed topological copy of $ mathbb S $. But $ F ( mathbb S) $ also contains a topological copy of the square $ mathbb S times mathbb S $ and so $ F ( mathbb S) $ contains an unobtrusive discrete subspace. Is this situation typical?

Problem. Let $ G $ to be a topological group containing a topological copy of the Sorgenfrey line. Is $ G $ does it necessarily contain an unobtrusive discrete subspace?

General Topology – What is the dimension of $ (C ([0,1], mathbb R), ||. || _ { infty} $?

After this question: A new generalization of dimension? part 2

I think there is a very interesting question: in the structure $ E = C ([0,1], mathbb R), ||. || _ infty $ with the closed set, this structure has a dimension, what is the dimension of E?

Result Super Bolzano: if $ A subset E $ and $ text {card} (A)> dim (E) $ then $ exists (a_n) _n in A ^ { mathbb N} $ injective with a limit in $ A $.

Perhaps it is easier to follow the path of structural theory.

Note: it's interesting only if $ dim (E) = text {card} ( mathbb N) $

at.algebraic topology – Generalize Wu's formula to general Bockstein homomorphisms

Wu's classic formula claims that
$$ Sq ^ 1 (x_ {d-1}) = w_1 (MC) cup x_ {d-1} $$
on a $ d $-collecteur $ M $, or $ x_ {d-1} in H ^ {d-1} (M, mathbb {Z} _2) $.

I wonder if there is a generalization of Wu's classical formula to general Bockstein homomorphisms. We consider Bockstein's homomorphism
$$ beta _ {(2,2 ^ n)}: H ^ * (-, mathbb {Z} _ {2 ^ n}) to H ^ {* + 1} (-, mathbb {Z} _2) $$
which is associated with the extension $ mathbb {Z} _2 to mathbb {Z} _ {2 ^ {n + 1}} to mathbb {Z} _ {2 ^ n} $.

I suppose that there is a generalized formula of Wu:
$$ boxed { beta _ {(2,2 ^ n)} (x_ {d-1}) = frac {1} {2 ^ {n-1}} tilde w_1 (TM) cup x_ { d -1}} $$
on a $ d $-collecteur $ M $, or $ x_ {d-1} in H ^ {d-1} (M, mathbb {Z} _ {2 ^ n}) $.

Right here $ tilde w_1 (TM) $ is the first twisted class of Stiefel-Whitney
the tangent package $ TM $ of $ M $ which is the withdrawal of $ tilde w_1 $ under the classification card $ M to BO (d) $.
Let $ mathbb {Z} _ {w_1} $ designates the local orientation system, the first class of Stiefel-Whitney twisted $ tilde w_1 in H ^ 1 (BO (d), mathbb {Z} _ {w_1}) $ is the removal of the non-zero element from $ H ^ 1 (BO (1), mathbb {Z} _ {w_1}) = mathbb {Z} _2 $ under the decisive map $ B det: BO (d) to BO (1) $.

The right side makes sense since $ 2 tilde w_1 (TM) = $ 0.

Can you help me prove or disprove the framed formula above?

Thank you!

What are the advantages and disadvantages of using enlargers for photography in general?

Magnifying lenses seem to be available at very little cost. What are the advantages and disadvantages of using enlargers for photography in general?

General Topology – Homotopy and "transfer" equivalence

Let $ X, Y $ to be topological spaces and $ f, g: X to Y $ to be continuous. We say that there is a transfer function of $ f $ at $ g $ if there is $ u: Y to Y $ such as $ g = u circ f $and we write $ g leq_t f $. We say that $ f, g $ are transfer equivalent if $ g leq_t f $ and $ f leq_t g $.

What's an example of spaces $ X, Y $ and continuous functions $ f, g: X to Y $ such as $ f, g $ are homotopic, but not equivalent by transfer – and what is the case for continuous functions that are equivalent by transfer, but not homotopic?

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General Mathematics – Mathematical Research in North Korea – Reference Request

Question: Where to find information on the areas of mathematics
are represented in which of the more than 20 universities of the
Democratic People's Republic of Korea (DPRK), and on which mathematicians
work there?

The DPRK is a country of about 25 million inhabitants.
and he is industrialized to a degree that allowed him to succeed
build nuclear weapons and ICBMs. So, we would expect there to be
a decent number of mathematicians working in his universities.

However, since the country has its own intranet, it is not very visible from there.
Open Internet. – Google will not help much further here.
In addition, most of the results of DPRK researchers are only published in
the country's national journals and mathematicians could not be found
in the mathematical genealogy database.

On the other hand, DPRK citizens who need the Internet to work
have access – of course with sites like Google or Facebook stranded as
to allow them to open all the doors of the CIA, the NSA, et cetera.
In addition, the exchange of emails between citizens of the DPRK and citizens of other countries
possible in the world (possibly monitored, for obvious reasons of
national security – keep in mind that the United States is still officially at war
with the country, and have recently even
threatened with total destruction).

As a remark, general impressions of the country can be obtained.
on Korea Central TV (TV shows on working days from approximately 06:00 UTC)
at 13:30 UTC, so from morning to the beginning of the afternoon to the European time
at night in American time zones and approximately in the
afternoon and evening in Asian time zones). Documentation
which aired on January 5, 2019 can be viewed here.

Important note: Please stay on the subject! – In particular it is do not
the place to discuss the policy. Anyone who would like to say something
who does not belong to this place is welcome to contact me privately by email

cross platform – Angular Concepts (2+) and XAML: search for general comparisons / contrasts, equivalent terminology, common traps, etc.

I am currently learning XAML / Xamarin.Forms and of course I am trying to relate its concepts to the framework I know best, namely Angular. Let me be clear that I understand that these two are very different and that they are targeting different platforms (although it seems that work is underway to bring Xamarin to the Web, which is exciting), but there are nevertheless obvious similarities, especially in their architecture. I think it would be really helpful to know how the two differ or when they do essentially the same thing, but with different terminology, but after some extensive research on the web, I'm almost empty.

I am enrolled in the software development program at my local technology school, which requires special attention to computer programming or web development (there is also a mobile solution, but this option is being phased out). I chose the office path and, although I am very comfortable with the .NET framework and C # coding, I only recently learned about XML and XAML ( only WPF) in the last two chapters of .NET II, ​​which is rather the last class of desktop-specific programming I will take.

On the other hand, all software development students (including desktop computers) must take Web I – III courses as well as cross – platform development, which are taught by a younger, more progressive teacher. That's why I'm a little familiar with Angular. and also ionic.

TL; DR Although I prefer to work in the Microsoft world, my ability to create more complex cross-platform applications using Angular / Ionic m incline towards web development. That's why I'm teaching XAML / Xamarin to fill this gap. I request links to resources (and of course all personal experiences) that show the relationships between the two and / or point out major differences, especially in their architecture (eg, models, data link, dependency injection). For those of you who have used both, I would like to hear what you like or not, or if you prefer something completely different from React.

Already, I see two negative votes, if this is not the place to ask this question, thank you for letting me know why and possibly lead me to another forum. I chose to post here because it seems more appropriate for more open questions than for something like StackOverflow.

General Topology – Compact Spaces and Urysohn Lemma

Let X be a metric space, A a non-empty subset of X and $ x in X $ a point. We define the distance d (x, A) via

$ d (x, A) = inf ({d (x, a) | a in A}) $

(a) Show that d: X → R is a continuous function and that x is in A if and only if $ d (x, A) = $ 0.

(b) Now leave A and B closed and non-empty subsets of X disjoint, and consider the function f: X → R
Defined by

## EQU1 ##

Show that f is well defined (that is, the denominator is never zero) and has the following properties:
in the lemma of Urysohn, that is to say that it takes value $[0, 1]$ and satisfied $ f (a) = $ 0 for all one ∈ A and $ f (b) = $ 1 for
all b ∈ B.


Show that in fact $ A = f ^ {- 1} (0) $

and $ B = f ^ {- 1} (1) $

My minds
can we say that the function is continuous if we have $ x, y in X $ so that

$ d (x, A) leq d (x, a) leq d (x, y) + d (y, a) $
for $ a in A $
$ d (x, A) -d (x, y) leq inf {d (y, a)} = d (y, A) $
$ d (x, A) -d (y, A) leq d (x, y) $

is it true or am I missing something? And how can I show that if $ x in overline {A} if and only if d (x, A) = 0 $
Is it fair to show that we can create a ball in A and show that the intersection is not empty?

I do not know how to prove b) and c)

Can someone help with that?

What is SEO in simple terms? – Help on SEO (general discussion)

SEO or Search Engine Optimization is the name given to the activity that attempts to improve search engine rankings. In search results, Google displays a link to pages that it considers relevant and authoritative. … In simple terms, your web pages have the potential to appear in Google ™ as long as other web pages refer to