Dimension, graph of functions of several variable and it's visualization. To visualise a scalar function of $n$ variables we consider its graph in $(n + 1)$ dimensional space.
If $\mathit{f} :U \subset \mathbb{R}^n \to \mathbb{R}$ is a function of n variable its graph consists of the set of points $(x_1,....,x_n,\mathit{f}(x_1,....,x_n))$ in $\mathbb{R}^{n+1}$ for $(x_1,....,x_n)$ in $U$
This lines are from my reference.
Considering the above definition.
I suppose, dimension of a set as the number of independent variables (i.e scalars) that is required to describe the position of a point of the set.(also considering for sets which are not vector spaces )
Now, 
$$\mathrm{S} =\lbrace (x,y,z)\in \mathbb{R}^3 | x^2+y^2+z^2=1\rbrace$$
(this set is not a vector space),now any point of this set can be represented using two scalars $(x,y,\pm \sqrt{1-x^2-y^2})$
So dimension of $\mathrm{S}=2$
$$\mathrm{T} =\lbrace (x,y,z)\in \mathbb{R}^3 | x^2+y^2+z^2\le1\rbrace$$
(also this set is not a vector space)
Clearly, the points of this set cannot be represented by $x$ and $y$ alone so we need 3 scalar $(x,y,z)$ to represent the element of this set $\mathrm{T}$.
So, dimension of $\mathrm{T}=3$
Physically, $\mathrm{S}$ is a sphere( or better to say ,here a spherical shell )
And $\mathrm{T}$ is a spherical ball.
So, from my perspective of definition of dimension. $\mathrm{S}$ is a $2D$ object and   $\mathrm{T}$  is a $3D$ object. Now if we plot this in any 3D grapher we certainly cannot distinguish between this two objects(graphs)
My questions are:


*

*Is my view for dimension correct? Is there anything like dimension for non vector space set?

*Are the sets that represent paraboloid, hyperboloid,or  a right circular cylinder all 2D objects?

*What is the difference between object and its graph?

*Considering the definition of my book do we need 4D space to visualise the set $\mathrm{T}$? And how it is different in visualization with the set $\mathrm{S}$
I am not sure of the tags i used.
 A: *

*Your view of dimension is correct. Indeed, the very first sentence on Wikipedia is what you say:


https://en.wikipedia.org/wiki/Dimension
There are more technical ways of defining it, but your intuition is correct.


*Yes, hyperboloid, paraboloid, cylinder are all 2-dimensional.

*I'm not sure what you mean by an "object".

*No, $T$ can be visualized in 3d space. It is the set of points inside the sphere. It is different from $S$ in that $S$ is only the points on the boundary of the sphere, but $T$ are all the points inside.
The major difference between $T$ and $S$ is that $S$ is defined by an equation, but $T$ is defined by an inequality. Generically, each equation needed to define a surface/manifold decreases the dimension by 1. For example:


*

*One equation in $(x,y)$ defines a curve (1-dimensional)

*One equation in $(x,y,z)$ defines a surface (2-dimensional)

*Two equations in $(x,y,z)$ defines a curve (1-dimensional)


On the other hand, an inequality does not generically decrease the dimension, as you can see with your $T$ example.
