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


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 be differentiable 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,

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


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

at.algebraic topology – Topological vs. Algebraic Intersection Forms

Let $ X $ to be a complex projective surface simply connected (so a real $ 4 $-collecteur). Let $ (H ^ 2 (X, mathbb Z) / mathrm {tors}, q_X) $, $ (A ^ 1 (X) / mathrm {tors}, q_X & # 39;) $ to be the corresponding networks in cohomology and intersection theory, where $ q_X $ and $ q_X $ are the forms of intersection (bilinear).

Is there an example of $ X $ for which the two networks are not equivalent? And if we restricted $ q_X $ at $ H ^ 2 (X, mathbb Z) cap H ^ {1,1} (X) $?

Also, what can be said about the cycle map $ A ^ 1 (X) to H ^ 2 (X, mathbb Z) cap H ^ {1,1} (X) subseteq H ^ 2 (X, mathbb Z) $ in that case? And if we take coefficients in $ mathbb Q $?

general topology – Continuity of a vector function $ f: X to mathbb {R ^ n} $, where $ X $ is a topological space.

Let $ X $ to be a topological space, and let $ f: X to mathbb {R ^ n} $ to be a vector function on $ X $. We define the $ k ^ {th} $ coordinating function of $ f $, which we refer to as $ f ^ k $, be $ p_k circ f $ or $ p_k: mathbb {R ^ n} to mathbb {R} $ is the $ k ^ {th} $ projection on $ mathbb {R ^ n} $. I've proved that if $ f $ is continuous, as is each coordinate function. However, I wonder if the opposite is true; which means that it is necessarily for $ f $ be continuous if each coordinate function is continuous ?.

I appreciate any help. Thank you in advance.

general topology – Link between the topological dimension and the Hammel dimension (algebraic) of a vector space

I was wondering if there was a connection between these two dimension definitions in the case of a topological vector space. In fact, I know that the topological dimension sometimes coincides with other notions of dimension.

In addition, are there any interesting results on TVS coverage properties (such as paracompact, refinements, etc.)?

Aggressive geometry – The quasi-separation as a topological condition of the scheme

Let $ X $ to be a ploy. Is it true that morphism $ X rightarrow mathrm {Spec} , mathbb {Z} $ is quasi-separate iff the intersection of two quasi-compact open subspaces of the underlying space of $ X $ is it quasi-compact (so, in particular, quasi-separation is a purely topological condition)? If the answer is positive, what is a published reference where this is proven in detail?

reference request – topological entropy of the logistic map $ f (x) = mu x (1-x) $, $ f:[0,1] at [0,1]$ for $ mu in (1,3) $

As stated in the question, I want to find the topological entropy of the logistic map on the interval $[0,1]$ for a "nice" parameter range $ mu $to know $ mu in ($ 1.3). I think the fact that $ f:[0,1] at [0,1]$ is a very important additional requirement here that simplifies things.

I've tried something, but I'm not sure it's the right way to tackle the problem, but I'll describe it here anyway.

I know a theorem that says $ h_ {top} (f) = h_ {top} (f | _ {NW (f)}) $, or $ NW (f) $ is the set of non-wandering points of $ f $so I wanted to find this set. By making a lot of small images, I concluded that for $ x notin {0,1 } $, we should have $ lim_ {n to infty} f ^ {n} (x) = 1- frac {1} { mu} $, which is the other fixed point of $ f $. Moreover, the convergence seems quite simple (that is to say that it gets closer to each iteration), so I had the impression that I should have $ NW (f) = {0, 1- frac {1} { mu} $.

To confirm this, Wikipedia says:

By modifying the parameter r, one observes the following behavior:

With r between 0 and 1, the population will eventually die, regardless of the initial population.
With r between 1 and 2, the population will rapidly approach the r - 1 / r value, regardless of the initial population.
With r between 2 and 3, the population will eventually approach the same r - 1 / r value, but will fluctuate around this value for some time first. The convergence rate is linear, except for r = 3, where it is extremely slow, less than linear (see Bifurcation memory).

However, I have not been able to find any evidence of these claims. Can any one show me how to prove that or give me a reference where the proof is clearly written?

Also, if there is an easier way to find the topological entropy (again, I insist on the fact $ f:[0,1] at [0,1]$; I've lost a lot of time reading about Mandelbrot sets by combining $ f $ at $ g (z) = z ^ 2 + c $ and looking at formulas for the entropy of $ g $ that exist but with domains $ mathbb {C} $ or a variant), I would be very happy to hear it.

ct.category Theory – A special monomorphism to encode the inclusion of topological subtopoids

Consider the category $ mathrm {TopMon} $ monoid topographies and continuous monoid homomorphisms.
Consider inclusion $ i: Bbb {R} _ { ge 0} hookrightarrow Bbb {R} $, where the spaces are taken with their usual topology and added as a monoid operation. This map is a monomorphism. Compare it now with the map (still injective) $ j: Bbb {R} _ { ge 0, D} hookrightarrow Bbb {R} $, or $ Bbb {R} _ { ge 0, D} $ is now the monoid $ ( Bbb {R} _ { ge 0}, +, 0) $ equipped with discreet topology.

I am looking for a formal way (theoretical category) to express this $ i $ is "more pleasant" than $ jsince for example $ i $ is the inclusion of a subspace, while $ j is not. however,
$ i $ is not strong mono nor extremal mono (since it is ear as well), nor mono regular, et cetera. Is there a less restrictive class than the strong or extremal mono to which $ i $ belongs but not $ j?

Of course, one possible way would be to take the obvious forgetful functor $ U: mathrm {TopMon} to mathrm {Top} $ and say that $ Ui $ is extremal mono, while $ UJ $ is not. However, I was wondering if there was a way to distinguish the two that is internal to $ mathrm {TopMon} $.

reference request – Topological invariants for group

Let $ mathbf {Grp} $ to be the category of groups and $ mathbf {Top} $ to be the category of topological spaces. To each group $ (G, circ_G) $we can associate a topological space $ (G, tau_G) $ the basis of this topology being given by all of the subgroups of $ G $. Call this topology on $ G $ to be his Subgroup topology So we get a functor $ mathscr {F}: mathbf {Grp} to mathbf {Top} $ which associates a given group with its. Also note that any homomorphism $ f: (G, circ_G) to (H, circ_H) $ induces a continuous function between the correspondent Subgroups of Topological Spaces.

This process looks like a kind of "reverse process" compared to what we do in algebraic topology, especially when we try to associate the fundamental group with a given topological space. In Algebraic Topology in general, we try to find algebraic invariants of a given topological space, while I try to find topological invariants for a group.

However, it is clear that the functor I defined above is only an example of a functor of $ mathbf {Grp} $ at $ mathbf {Top} $ and (I think) is not going to be very useful.

So my question is,

Is there a useful topological invariant of a group? More specifically, since any group can be associated with a topological space (as we did for the fundamental groups)? If yes, can we talk about literature?

lo.logic – Examples of topological spaces Kreisel-Putnam

Let's say that a topological space $ X $ is a Kreisel-Putnam space when it satisfies the following property:

For all open sets $ V_1, V_2 $ and regular open set $ W $ of $ X $, if a point $ x in X $ has a neighborhood $ N $ such as $ N cap W subseteq V_1 cup V_2 $ so in fact he has a neighborhood $ N & # 39; as is $ N / # cape subseteq $ V_1 or $ N cap subseteq V_2 $. (Equivalently, $ operatorname {int} (V_1 cup V_2 cup (X setminus W)) $ is contained in the union of $ operatorname {int} (V_1 cup (X setminus W)) $ and $ operatorname {int} (V_2 cup (X setminus W)) $.)

(There may be some clearer ways to formulate this, perhaps switching to closed complements is more acceptable.)

The reason for this name and condition is that the foregoing equates to saying that the Heyting algebra of open sets of $ X $ satisfies Kreisel-Putnam axiom $ ( neg u Rightarrow (v_1 lor v_2)) Rightarrow (( u u Rightarrow v_1) lor ( neg u Rightarrow v_2)) $ of interest for the study of intermediate logics. But of course, if this property has a different, more classic name, that's part of my question.

A counter-example is provided by $ mathbb {R} ^ 2 $: this does not satisfy the Kreisel-Putnam property, as shown $ V_1 = {x_1> 0 } $ and $ V_2 = {x_2> 0 } $ and $ W = V_1 cup V_2 $, which is indeed regularly open, and the point $ x = (0,0) $. (A counter-example a little more complicated works for $ mathbb {R} $.)

Note the requirement that $ W $ be regular open (or equivalent, be the pseudocomplement $ operatorname {int} (X setminus U) $ of an open set $ U $). If we abandon this requirement, we get (probably!) A stronger condition on $ X $ that I could call a Gödel-Dummett space, because his open sets Heyting algebra satisfies the axiom $ (w Rightarrow (v_1 lor_v_2)) Rightarrow ((w Rightarrow v_1) lor (w Rightarrow v_2)) $, which turns out to be equivalent to $ (v_1 Rightarrow v_2) lor (v_2 Rightarrow v_1) $, the Gödel-Dummett axiom.

As I find this property a bit difficult to visualize, I would like to ask:

Question: What are some interesting examples of Kreisel-Putnam spaces?

I do not have a precise definition of "interesting", of course (I'm trying to grasp the notion intuitively), but, for example, discrete spaces (which are actually Kreisel-Putnam's ) are definitely do not interesting. Ideally, I'd like something that looks like "a little bit like $ mathbb {R} ^ n $"But some of the criteria that could help make an interesting space might be: be regular, connected (or at least not extremely disconnected) and do not satisfying the Gödel-Dummett condition.

It should probably be pointed out that, as this answer shows, the toposs of simplicial sets satisfy the Kreisel-Putnam axiom. The corresponding condition for the topos of sets of sets on $ X $ would be that each set open in $ X $ satisfied with the Kreisel-Putnam condition (perhaps this simply stems from $ X $ satisfactory, this is one of the many things that are not clear to me).

Functional Analysis – Is the continuous double of a topological chain complex equivalent to a chain of a topological chain complex?

I apologize in advance if it's a naive question.

def: A topological chain complex is a chain complex of $ mathbb {R} $– Vector spaces such as boundary maps are continuous.

Let $ C $ to be a topological chain complex.

we can naturally consider the double $ C $* of the complex of the chain $ C $ (ignoring the topology).

We can also consider the continued double the complex of the topological chain $ C $** (because the boundary maps of $ C $ are continuous)

Are $ C $* $ C $** Chain equivalent? I would also like to know if the answer is yes after possibly adding some additional conditions to avoid any risk of pathology.

Thank you,