Dimensional Tiering

== N-Dimensional Spaces ==
It is important to understand what a unit interval is in mathematics. A unit interval is an open interval between 0 and 1, it is a subset of real numbers. Because of the nature of real numbers, specifically how they can be used to represent points along an infinitely long line. The line segment of length 1, which is practically the unit interval in this case is a part of the infinitely long line that real numbers represent points along and has the same number of points as the whole line. 
This means the unit interval or open interval between 0 and 1 is equivalent to a set of real numbers entirely. This is why it's regarded as an uncountable set. Now how is this significant for dimensions? Because dimensions are n-tuples of real numbers. Typically, each coordinate axis of a dimension is an equivalent of the unit interval, it then follows that every given physical space is an equivalent to a set of real numbers.
For any given n-dimensional space, an n-dimensional is continuous, a geometric space \Re^n is referred to as an n-dimensional space with n spatial dimensions, wherein n can be any given positive integer, in this sense \Re^1 then persists as an infinite line, \Re^2 an infinite plane and \Re^3 persists as a three-dimensional space that is infinite in all directions (of which we know of).
\Re^n being referred to as space is a way in which mathematicians extend their geometric intuition for lower-dimensional spaces to higher dimensions. For any given point in n dimensions, it follows that it is a list of real numbers, take an example with\{3,4\} being such that it represents a point in 2-dimensions. Analogously \{3,4,8\}would then represent a point in a 3-dimensional space; the coordinates typically being called x, y and z.
==Higher Dimensions==
When n\geq4, it is said that the space is higher-dimensional, as a result of every dimension being over \Re, every dimension is of the same order of infinity, but what defines a higher-dimensional space as ontic status beyond lower-dimensional space is not cardinality rather other topological elements and configurations.
Suppose we have \Re^2 with coordinates x and y, z=0 defines the x, y-plane which stands in relation to \Re^2 as a result of the points on the given plane having the z-coordinate that is zero. Analogously if we have an n-cube or hypercube being defined by a set of points \{x1, x2, x3,......., xn\} is such that any coordinate xi ranges from 0 to 1. 
Whereby 0 and 1 is a unit interval and in relation to n-dimensional spaces, coordinates of points lie between 0 and 1, each one dimension is then defined as a unit interval in itself, it then follows that each coordinate xi, whereby i ranges from 1 to n\{x1, x2, x3,......., xn\}, the values of i are constrained to the interval [0,1]. It then follows that for any given dimension, suppose its \Re^4 defined as a hypercube, one of its faces can be ascertained by fixing one coordinate (in this case a coordinate likex1=0, which creates a subspace \Re^3 inside of \Re^4 and reduces the dimension of the given space).
If the face is defined by x1=1 corresponds to a set of points \{1, x2, x3, x4\} wherein 0< x2, x3, x40,<x1,<x2,<x3,<x4,<1. The fact is said to be a 3-dimensional cube. The parallel face at x1=0 represents another 3-cube described by the points\{0, x2, x3, x4\} and the condition x1=0 fixes the first coordinate to 0. In this case x2,x3,x4 then independently take any value from 0 to 1. The 3-cube then will persist within 4-dimensional space of the hypercube and one of its faces, suppose its x1=1 for example is just another 3-cube.
This intuition extends to any dimension n, illustrating exactly how each higher-dimensional space contains multiple lower-dimensional faces, all of which can be examined by fixing coordinates. A higher dimension despite being of the same order of infinity as lower dimensions, exhibits greater complexity than its lower-dimensional equivalents, each point in a lower-dimensional space corresponds to multiple configurations of the higher-dimensional space itself.
Each n\geq4 space is an extension of a lower dimensional space, embedding itself onto lower dimensions, with each additional dimension arising greater degrees of freedom, allowing for even more complex configurations which are otherwise impossible or don't persist within lower dimensions. Points of \Re^1 despite being infinitely many points,\Re^2 has an even greater infinity of points encompassing all possible configurations and combinations of the given points prior to and beyond or more.
== Temporal Dimensions==
Discreteness of time t implies time t is finitely divisible, insofar as t is divided into purely distinct intervals, where each interval can be equally associated some index n. tn: tn then represents any n-th discrete interval, where n is an integer that specifies each point or interval t0, t1, t2, t3, t4........ tn, whereby t0 is the starting point and t1 is the successive point of t0.
====With the sequence notation being====
\sum_{n=0}^N
Would represent the entire sequence of discrete time from t0 to tn, which can be alternatively represented as just t= {t0, t1, t2, t3.... tn} being the entire set of the sequence of discrete time. The intervals between the successive time points then remain constant or the difference between consecutive time indices each correspond to a fixed duration in real time, as a result of time progressing uniformly in a discrete-time system.
Formally speaking the symmetry breaker between any consecutive time points tn and tn+1 remain constant, with the difference being outlined or denoted as t. Then if tn is the time at the n-th step then it logically follows that
tn+1 = tn+\bigtriangleup t
then tn=t0 + n\bigtriangleup t
In simple terms, If t= 1 second, then t0 = 0 seconds, t1 = 1 second, t2 = 2 seconds,
Common ways to represent continuous time with the variable t, is t\in\Re
, whereby T can be any real number representing continuous time. With the interval notation (In) being such that,IN=t\in[t0,tf]
and t\in[0,15]
 Can represent continuous time from 0 to 15 seconds and CTS={\operatorname{dx}\!(t)\over\operatorname{d}\!x}
Represents changes over time t. Unlike discrete time, time progresses smoothly and can take any value within a space-time continuum, there are no real gaps between temporal points, functions in discrete time are defined only at discrete points, a discrete sequence for example being represented as x[n], where n is an integer index.
Differs from a continuous one, whereby continuous time is defined at every instant, that is to say, time can take any value in continuous range, a state of a system in itself can be described at every possible moment in that range, between two moments in time, there are infinitely many other moments in it. For example, if you consider two times, say t1= 1 second and t2 = 2 seconds, there are infinitely many points in between these two, such as 1.01 seconds, 1.001 seconds and so on.
==Metric Expansion==
To address a hypothetical expanding universe within a given franchise, now if it is the case the universe expands in a given franchise. It is possible for it to expand without compromising space being infinite and does not necessarily imply it being a literal expansion of the spatial dimensions or the universe getting bigger in a literal sense.
It instead is like an infinite grid of points, in which the distance between each point increases rather than the infinite grid in itself growing. In this case the expansion happens but space in itself is infinite. A universe is either Flat Universe, in which it being flat means it follows the rules of Euclidean Geometry, which is over reals, a flat universe is infinite insofar as it has no boundaries and extends infinitely in all directions, Baring the Friedmann-Lemaitre-Robertson-walker metric:
ds^2=c^2dt^2+a(t)^2(dx^2+dy^2+dz^2)
*(ds) is thee space-time interval
*(c) is the speed of light
* (t) is a time coordinate
*(a(t)) is a scale factor describing how the distance in the universe expands over time 
*(x, y, z) all being spatial coordinates
The metric expansion can occur, but it doesn't imply discreteness for the universe. Despite being infinite, space then expands insofar as the scale factor a(t)
 increases over time, which suggests that the physical distance d(t)
 between any two points in the universe increases as:
d(t)=a(t)\cdot d0
Whereby d0
 signifies the distance between points measured when the scale factora(t)=1
Comoving Coordinates are best imagined as a grid, whereby as the universe expands, the Comoving distance between two points on the grid doesn't change in itself, even when the proper distance between them increases over time as the universe expands.
Assume we have two galaxies, that are 2 billion light-years apart or so, if the universe expands such that the scale factor doubles, their proper distance then would be 3 billion light-years apart, the Comoving distance between the two of them will remain at 2 billion light years, even when the proper distance of them changes, the Comoving distance remains fixed.
And finally, a closed universe, as a I explained a while back? A finite and bounded universe has finite size, it still has no boundaries and edges, but if one where to hypothetically travel in a straight line, you'd eventually return to your stating point, typically the universe expanding means at some point it will eventually collapse back into the singularity or contract after.
A possible metric that can be used to describe it is
ds^2=\Re^2[d\psi^2-d\chi^2-\sin^2\chi d\Omega ^2]
The presence of\sin^2\Chi
 being indicative of a closed geometry, \Chi
 ranges from 0 to \pi
and the sine function describes the sphere-like surface where spatial dimensions are curved eventually looping back on by themselves (hence are of finite extent). \Psi
 being a temporal-like coordinate \chi
 a spatial-lie coordinate, R being the radius of the universe. A close and bounded universe in this case can be described by coordinates \Psi
 and \Psi
 with ranges:
0<\Psi<2\pi, 0<\Chi<\pi
Which accounts for its bounded nature. These finite ranges imply that the universe is limited in a spatio-temporal sense, in this the closed universe has no boundaries but it is finite in size similar to the surface of a sphere.
An open universe has a hyperbolic geometrical structure, meaning it curves outward, it has no boundaries and extends infinitely as well, described by the FLRW metric with a different spatial part the metric expansion increases over time it also doesn't imply discreteness. 
Category:Standards for tiering
Category:WIP