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Is the function $F$, as defined below, possible? If yes, is there is a name\reference for such a function.

Function $F(X) = \{y_1, y_2, \ldots,y_n\}$ such that

  1. $X$ is a set (in my case, $X$ is a solid- a set of points)
  2. $y_i$ is also a set (in my case, $y_i$ is a solid, surface, or curve)
  3. $F$ is a group of functions $\{f_i (x_i) = y_i\}$ where $\{x_1, x_2,\ldots, x_n\}$ is some partition of $X$.

enter image description here

So, essentially, $F(X)$ maps sub-solids of solid $X$ into solids, surfaces, and curves.

I am familiar with the concept of restriction of a function, but it seems that it does not apply here.

Thank you!

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    Your description is somewhat hard to follow. I think it would help if you gave a more concrete example. – 2017-02-03
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    I included an illustration. – 2017-02-03
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    I would much rather see a concrete example. – 2017-02-04
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    I am not sure what do you mean by concrete? – 2017-02-04
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    A concrete example of addition would be $25+17=42$. A concrete example of your function would be stating an example of a concrete (exactly described) input and exactly what the output for that input should be – 2017-02-04
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    Then the example I gave is concrete in that sense. X is a solid bar (a point set) broken into three pieces. F is a function which transforms each piece of the bar into some other point set. I don't know what exactly the function looks like. I just have to say it exists. Sorry, I don't have any other way to put it. This is the exact type of transformation I want and was wondering if I can formally define it as a function. – 2017-02-04
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    What should your function return if the input is your bar minus one point in the interior? – 2017-02-04
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    It is still unclear even if you're talking about _one particular function_ that you just can define clearly, or about a general _kind of functions_. – 2017-02-04
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    Kind of function: In the example, the function's argument is a bar of some length and cross-section. Another function could be one whose domain is plate-like solids and outputs a bunch of surfaces, for example. – 2017-02-04
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    What would your example function return if the input is an open ball of radius 3 centered on $(0,0,0)$ together with the point on the north pole? – 2017-02-04
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/53039/discussion-between-od08-and-henning-makholm). – 2017-02-04

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