Name of a tuple of a set and a function $(X,f)$ akin to "topological space" or "metric space".

Question

Let $X$ be a set, $\tau$ a topology, $d$ a metric, and $f$ a function. Then we can construct a topological space $(X,\tau)$, where $\tau$ is a set of subsets of $X$. We can construct a metric space $(X,d)$ where $d$ is defined between each pair of elements of $X$. And we can construct a ??? space $(X,f)$, where $f$ is defined for each element of $X$...

Wait... what do we call the last thing? Sure, $X$ is the domain of $f$, but what is can we call $(X, f)$?

Background

I have a function $g: o \mapsto (X,f)$ and I would like to write a sentence, where I call $o$ an "object", like "each object needs to be mappable to a ??? space". I am also not sure if "mappable" is right here, or whether "morphable" or "functionable" is right. What I mean is that "there must exist a function that maps $o$ to $(X,f)$". Of course $o\not\in X$.

Ideas

Would I be allowed to call it a "function space" or "functional space" -- are these terms already taken for other things? I would of course define the new terminology in the "Notations" section.

Alternative ideas: Call it a "weight space", or "weightical space" (since $f$ is weighting each element of $X$). I am sure these are not taken yet... But maybe they sound weird...

Answer 1

Gae S is correct. If you go a bit further and talk about $(X, Y, f)$ (or whichever order you choose to put them in), where $f: X \to Y$, then there is indeed a special name for these objects. And just like we normally denote $(X, \tau)$ or $(X, d)$ by just $X$, suppressing the explicit mention of the topology or metric, we also normally denote $(X, Y, f)$ by just one symbol, leaving the other two implicit.

What we call $(X, Y, f)$ is a "function", and it is $X$ and $Y$ that are left implicit. But the full definition of a function requires specifying all three. And just like topologies and other spaces, this is done by a tuple. Since the cases you are interested in all have the same codomain, you are also ok in suppressing that element and just talking about $(X,f)$, provided your readers understand what the codomain is.

Now it may be that for your purposes, you really want to think about $(X, f)$ as a space of points $X$ endowed with $f$ as a property, rather than thinking of it as a function with domain $X$. That is, to emphasize the set of points rather than the functional relation. In this case, "Weighted space" might be appropriate. But really what terminology is best would depend on why you want to think of it as a space-with-property instead of a function. What are you using $f$ for? What does it model in your situation? The answer to those questions would be the best choice for naming the space.

Answer 2

The thing you are looking for is a fuzzy set:

A fuzzy set is a pair $(U,m)$ where $U$ is a set and $m\colon > U\rightarrow [0,1]$ a membership function. The reference set $U$ (sometimes denoted by $\Omega$ or $X$) is called universe of discourse, and for each $x\in U$, the value $m(x)$ is called the grade of membership of $x$ in $(U,m)$. The function $m=\mu _{A}$ is called the membership function of the fuzzy set $A=(U,m)$.

For a finite set $U=\{x_{1},\dots ,x_{n}\}$, the fuzzy set $(U,m)$ is often denoted by $\{m(x_{1})/x_{1},\dots ,m(x_{n})/x_{n}\}$.

Let $x\in U$. Then $x$ is called

• not included in the fuzzy set $(U,m)$ if $m(x)=0$ (no member),
• fully included if $m(x)=1$ (full member),
• partially included if $0<m(x)<1$ (fuzzy member).

The (crisp) set of all fuzzy sets on a universe $U$ is denoted with $SF(U)$ or sometimes just $F(U)$.

Here, $f$ / $m$ is not an arbitrary functions, but since you mentioned that weighting is your intention, this should be what you are searching for.