Timeline for Differentiable maps between topological spaces [closed]
Current License: CC BY-SA 3.0
28 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 16, 2013 at 3:32 | comment | added | Abdullah Almariah | I think it is impossible, but why? | |
| Dec 16, 2013 at 3:29 | comment | added | Abdullah Almariah | Why is the question closed, although no one can answer it? Using topological structure we can define continuous functions and convergent sequences. Why differentiable functions could not be defined in terms of the topology? | |
| Dec 15, 2013 at 6:43 | comment | added | Ali Taghavi | $C(X)$ is the space of all continuous function from X to $\mathbb{C}$. This is a $C^{*}$ algebra. | |
| Dec 14, 2013 at 21:28 | comment | added | Abdullah Almariah | Ali Taghavi: What do you mean with complex valued continuous functions on $X$ and $Y$, whereas you have defined $X$ and $Y$ as topological spaces? | |
| Dec 14, 2013 at 20:54 | comment | added | Ali Taghavi | Perhaps the following algebraic formulation could work::Let $X$ and $Y$ be two compact topological space and $A=C(X)$ and $B=C(Y)$ be the $C^{*}$ algebra of complex valued continuoes functions on $X$ and $Y$ resp.Choose (and fix) two dense subalgebra $A'$ of $A$ and $B'$ of $B$. every continuous function on $f:X \rightarrow Y$ define a natural morphism $f^{*}$from $B$ to $A$ we can say that $f$ is $A'-B'$ differentiable if $f^{*}(B) \subset A$. This could be considered as a natural generalization of standard differentability for maps between manifold. | |
| Dec 14, 2013 at 20:23 | comment | added | Joseph Van Name | I currently have no analogous idea for maps $f,g:X\rightarrow Y$ between topological spaces, and I doubt that it is even possible without adding extra structure to the topological spaces. | |
| Dec 14, 2013 at 19:21 | comment | added | Abdullah Almariah | Joseph Van Name: Do you have any analogous random idea for $f,g : X \to Y$, where $X$ and $Y$ are topological spaces? | |
| Dec 14, 2013 at 19:17 | comment | added | Abdullah Almariah | Joseph Van Name: You are right. I got confused because you have defined a function between a topological space and $\mathbb{R}$. | |
| Dec 14, 2013 at 19:00 | comment | added | Joseph Van Name | Abdullah Almariah. The notion of a limit makes sense general topological spaces. To make my definition more precise, $^{\lim}_{x\in U,x\rightarrow x_{0}}\frac{g(x)-g(x_{0})}{f(x)-f(x_{0})}=L$ if for each $\epsilon>0$ there is an open neighborhood $O$ of $x_{0}$ where if $x\in O\cap U$, then $|\frac{g(x)-g(x_{0})}{f(x)-f(x_{0})}-L|<\epsilon$. | |
| Dec 14, 2013 at 18:22 | comment | added | Abdullah Almariah | Joseph Van Name: you should define a metric on $X$ at first for you idea to be right or have a meaning. I am not sure. This like the answer of Todd Trimble, on metric space. | |
| Dec 14, 2013 at 17:54 | comment | added | Joseph Van Name | Here is a completely random idea. Suppose that $X$ is a topological space without isolated points and $K=\mathbb{R}$ or $K=\mathbb{C}$, and $f,g:X\rightarrow K$. Then we say that $\frac{dg}{df}(x_{0})$ exists if there is a dense open set $U\subseteq X$ where $\frac{dg}{df}(x_{0})=^{\lim}_{x\in U,x\rightarrow x_{0}}\frac{g(x)-g(x_{0})}{f(x)-f(x_{0})}$ exists. | |
| Dec 14, 2013 at 17:35 | comment | added | Abdullah Almariah | If you have any suggestion for improving the question, please post it! | |
| Dec 14, 2013 at 17:33 | vote | accept | Abdullah Almariah | ||
| Dec 14, 2013 at 17:33 | |||||
| Dec 14, 2013 at 17:30 | review | Reopen votes | |||
| Dec 14, 2013 at 22:07 | |||||
| Dec 14, 2013 at 17:18 | comment | added | Qfwfq | After reading the answer by Todd Trimble, I voted for reopening. But the question should be improved. @Almariah: could you provide some context? From where did you get the idea that a surrogate of the notion of differentiable maps could be available for (some classes of?) topological spaces? .... | |
| Dec 14, 2013 at 15:55 | history | closed |
Steven Landsburg Ryan Budney Benoît Kloeckner Chris Godsil Jack Huizenga |
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| Dec 14, 2013 at 15:53 | answer | added | Todd Trimble | timeline score: 2 | |
| Dec 14, 2013 at 15:15 | comment | added | Abdullah Almariah | What do you mean exactly with "goal"? Please give examples. | |
| Dec 14, 2013 at 15:12 | comment | added | Benoît Kloeckner | The very notion of differentiability involves measuring distances at least to some order (think Taylor formula), so I do not see that one could expect anything interesting with only a topological structure. But unless you precisely state what is goal, this is not a question that can be answered. | |
| Dec 14, 2013 at 15:08 | comment | added | Abdullah Almariah | you can say that differential topology is dealing with differentiable functions on differentiable manifolds. My question is dealing with differentiable functions on topological space in general. Is there an analogy in general topological space without any added structure or restrictions? | |
| Dec 14, 2013 at 15:03 | comment | added | Abdullah Almariah | en.wikipedia.org/wiki/Generalizations_of_the_derivative | |
| Dec 14, 2013 at 14:58 | comment | added | Ryan Budney | What properties would you like the definition to satisfy ? | |
| S Dec 14, 2013 at 14:54 | review | First posts | |||
| Dec 14, 2013 at 15:31 | |||||
| S Dec 14, 2013 at 14:54 | review | Close votes | |||
| Dec 14, 2013 at 15:55 | |||||
| Dec 14, 2013 at 14:47 | comment | added | Abdullah Almariah | I just mean the definition. Is there any definition of differentiability of maps between topological space. | |
| Dec 14, 2013 at 14:45 | comment | added | Ryan Budney | You can define anything you like, I suppose. It's especially easy if the object you want to define has no role to play. | |
| Dec 14, 2013 at 14:42 | comment | added | Patrick I-Z | You must be more precise in what you are willing to do. The answer to your question is depending on. | |
| Dec 14, 2013 at 14:37 | history | asked | Abdullah Almariah | CC BY-SA 3.0 |