[Mathematical Spaces]

• Spaces are a fundamental concept in Mathematics.
• There’re tons of spaces out there, with each built for its own purpose.
• topological Spaces: sets with a notion of what is "open" and "closed".
• Vector Spaces: sets with operations of "addition" and "(scalar) multiplication".
• Topological Vector Spaces: "Addition" and "multiplication" are continuous in the topology.
• Metric Spaces: sets that come with a way to measure the "distance" between two points, called a metric; the topology is generated by this metric.
• Locally Convex Spaces: sets where the topology is generated by translations of "balls" (balanced, absorbent, convex sets); do not necessarily have a notion of "distance".
• Normed Vector Spaces: sets where the topology is generated by a norm, which in some sense is the measure of a vector's "length". A norm can always generate a metric (measure the "length" of the difference between two vectors), and every normed space is also locally convex.
• Fréchet Spaces: a set where the topology is generated by a translation-invariant metric; this metric doesn't necessarily have to come from a norm. All Fréchet spaces are complete metric spaces (meaning that if elements of a sequence get arbitrarily "close", then the sequence must converge to an element already in the space.)
• Banach Spaces: a set that is a complete metric space, where the metric is defined in terms of a norm.
• Inner Product Spaces: sets with a way to measure "angles" between vectors, called an inner product. An inner product can always generate a norm, but the space may or may not be complete with respect to this norm.
• Hilbert Spaces: an inner product space that is complete with respect to this induced norm. Any inner product space that is incomplete (called a "pre-Hilbert Space") can be completed to a Hilbert space.
• Manifold: a set with a topology that locally "looks like" Euclidean space. Any manifold can be turned into a metric space.

https://www.localmaxradio.com/questions/what-is-a-mathematical-space

https://www.math.toronto.edu/gscott/WhatVS.pdf

https://www.cuemath.com/geometry/

• It defines the notion of open sets and specifies which sets are considered open, allowing us to define concepts such as continuity, convergence, and connectedness.

Would it be safe to make the following generalization? topological space--->metric space---->euclidean space

Reference: https://www.physicsforums.com/threads/topological-space-euclidean-space-and-metric-space-what-are-the-difference.416148/

yes. Certain spatial properties of Euclidean space are abstracted to get the notion of a topological space. metric spaces are in-between the two, they are a special kind of topological space, but there are several possible metrics on a given set, including R^n. Of these, only one is the standard Euclidean metric on R^n: d(x,y) = √()

Reference: https://www.physicsforums.com/threads/topological-space-euclidean-space-and-metric-space-what-are-the-difference.416148/

Euclidean 𝑛�-space is (most commonly) the space ℝ𝑛��, with the dot product giving an inner product. This makes Euclidean 𝑛�-space into an inner product space, which means it is a normed vector space, hence has a topological structure. In summary, a topological space is a huge generalization of Euclidean space, and all the notions from topology apply.

Euclidean space is a vector space over ℝ� with an inner product.

You have the following inductions going on.

An inner product induces Norm induces Metric induces Topology

So Euclidean space definitely is a topological space and even a so-called topological vector space.

Topological Space

• A topological space is a mathematical concept that provides a framework for studying properties of sets and their relationships, specifically focusing on the notion of "closeness" and "openness”
• All spaces inherit from Topological space.
• It’s the root of the hierarchy of mathematical spaces. (Similar to the base object, the root of animals, vehicles, and buildings in Programming)
• Topological space doesn't make use of any geometric definitions (Graphical Drawings).
• It generalizes many of the ideas that were first developed in the Euclidean Space.
• Topological space is only represented using the language of the set theory.
  • It is a mathematical concept that provides a framework for studying the properties of sets and their relationships.
  • We typically focus here on the notion of closeness and openness.
• The topological space as mentioned is not graphical but represented in sets ———> We define the Topological Space as $(X, \tau)$.
  • $X$ is the collection of all of the points we have.
  • $\tau$ is a family of subsets of set $X$ —— We call $\tau$ the topology —— We call these subsets OPEN SETS.
    • Note that $\tau$ doesn’t have all subsets of X. Only in an extreme case, the discrete topology, the $\tau$ will be the power set.
  • The pair $(X, \tau)$ is called the topological space.

• In order to consider that pair a topological space, the collection of subsets must satisfy three properties:
  1. The empty set (∅) and the whole set X are both open sets.
     • Which means they must be in $\tau$
  2. The intersection of any finite number of open sets is also an open set.
     • In other words: $\tau$ is closed under finite intersections.
  3. Any union of any subset of $\tau$ must be in $\tau$.
     • In other words: The union of any collection of open sets is also an open set.
     • In other words: $\tau$ is closed under arbitrary unions.
     • Union is different from intersections because we don’t require the # of unions to be finite.
     • We can do up to an infinite number of unions, it still must be an open set.

• Which one of the following are topologies?
  1. The first one satisfies all 3 properties
  2. The intersection of {1,2} and {2,3} gives {2} and not in T — Not Topology
  3. The union {1,2} and {2,3} is not in T —— Not Topology
  4. It satisfies all 3 properties —— T is a topology here

• Indiscrete Topology OR Trivial Topology
  • Native Example of a topological space: $\tau = (\phi, X)$
  • It is the coarsest topology possible on a space that satisfies the axioms, and hence it is always worth having in your mind.
  • Rule #1 is already satisfied literally.
  • $\phi \cup \phi = \phi$, $\phi \cup X = X$, $X \cup X = X$ —— #2
  • $\phi \cap \phi = \phi$, $\phi \cap X = \phi$, $X \cap X = X$ —— #3
  • This is considered the minimal topology —— Because a lot of subsets of X are not an open set —— But remember that the condition of the topological space focuses on $\tau$ , not $X$.
  • The indiscrete topology is always “included” in any topology you consider.
• Discrete Topology
  • It’s a particular type of Topological Space.
  • The topology $(\tau)$ consists of all possible subsets of the underlying set. ——> $\tau = \mathcal{P} (X)$ OR $\tau = \{{\text{all subsets of }X}\}$
  • Here, the topology is the power set of the underlying set, denoted as $\mathcal{P} (X)$
  • Here, every subset of X is an open set.
  • Ex: Any point of X that could be a singleton set is an open set. Such as S = {1}
  • Ex: any combination of points on the entire set X itself is also open.
  • Ex: If we have the set X of {a, b, c}, then our topology is $\phi$, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, and {a,b,c}
  • The discrete topology with the power set as the topology is an extreme example where every possible collection of points is considered open.
  • This is considered the maximal topology —— Because every subset here of $X$ is an open set —— But remember that the condition of the topological space focuses on $\tau$ , not $X$.

Closure of a Set

What is a neighborhood?

• That’s a very very easy term to understand.
• A neighborhood of a point is a set that contains the point itself as well as some nearby points.
• Example 1: The set of real numbers ℝ. If we choose a specific point, say 2, then a neighborhood of 2 could be represented by an interval around 2, such as (1, 3).
• Example 2: The same, if we consider a set of points in a plane, let's say {(1, 1), (2, 2), (3, 3)}, a neighborhood of the point (2, 2) could be represented by a circle centered at (2, 2) with a certain radius, say 1 unit. This neighborhood would include all the points within a distance of 1 unit from (2, 2), including (1, 1) and (3, 3).

What is a closed set?

• The set $X \in R$ is said to be closed if it contains all of its limit points.
• In the context of topology, closed sets are not considered as fundamental as open sets.
• Open sets are often used as a starting point to define a topology, and the concept of closed sets arises as the complementary notion to open sets.
• Closed sets play an essential role in various areas of topology, such as closure, limit points, and continuity of functions.

What are limit points?

• Imagine you have a set of points, let's call it set A. A limit point of set A is a point that can be very close to A, even if it's not exactly in that open set.
• Example: if set A is a line segment. The endpoints of the line segment are limit points because you can approach them closely from either direction along the line.
  • They are not part of the line segment, but they are very close to it.
• The boundary of a set A, denoted as ∂A, is defined as the set of points that are both in the closure of $A$ and in the closure of the complement of $A$.
  • That means these limit points close A, and close A complement as well.
  • In other words, the boundary consists of points that are "close" to both $A$ and its complement.

Is the intersection of an open set with a closed set open, closed, or neither?

What is the closure of a set?

• It’s like we have an open set, and we want to close it نختتمها, so we find closure for it.
• The closure of A is the smallest closed set that contains A.
• The closure of a subset S of points in a topological space consists of all points in S together with all limit points of S.
• In simple terms, we have a subset of our all points collection X. This subset has points inside it, and has a boundary (limit points) —> The closure = The union of S and its boundary.
• In simpler words, all the points are either in S or "very near" S.
• In topology, we find the closure of an open set in these steps:
  1. We have our topology $(X, \tau)$, and we have our own open set A which is in $\tau$.
  2. We can easily get all close sets by getting the complement of every set in $\tau$.
  3. Now, if you want to find the closure of set A, you want the minimal boundary points that can make it close.
  4. We get all closed sets that have already A as part of it. Then we intersect to get the minimal closed set.
  5. WE ARE DONE! VERY INTERESTING APPROACH!
• Cl(S) is a closed superset of S.
• Explaining in terms of Neighborhood Terms:
  • We said a neighborhood of a point is a set that contains the point itself as well as some nearby points.
  • Consider a set S and its closure cl(S).
  • If we take a point p outside the set S, but p has a neighborhood that intersects with S, then p is a limit point of S.

Example (0): Set S = {0, 1, 2}. What is the closure of (S)?

Example (1): X = $R$, A = (a,b), what’s the closure of A?

• Simply without any topology, The closure will be $\text{cl}\{A\} = [a,b]$ because those are the closest points to the open set.

• A closed interval is a closed set.
  • [b, c] is a closed interval means all real numbers between b and c, including b and c.

• Standard Topology
  • $X = R^1$ —— Where $R^1$ is the real line numbers

I had underestimated the power of the English language to suggest mathematically incorrect statements to my students. In mathematics, "open" and "closed" are not antonyms. Sets can be open, closed, both, or neither. (A set that is both open and closed is sometimes called "clopen.") The definition of "closed" involves some amount of "opposite-ness," in that the complement of a set is kind of its "opposite," but closed and open themselves are not opposites. If a set is not open, that doesn't make it closed, and if a set is closed, that doesn't mean it can't be open. They're related, but it's not a mutually exclusive relationship. But in English, the two words are basically opposites (although for doors and lids, we have the option of "ajar" in addition to open and closed). My students used their intuition about the way the words "open" and "closed" relate to each other in English and applied that intuition to the mathematical use of the terms.

Metric Spaces

• Metric space is a set X that has a notion of the distance $d(x, y)$ between every pair of points x, y ∈ X.
  • $X$ is a set of different points, where between any pair of those points, we can calculate the distance.
  • The distance is measured by a function called a metric or distance function.
• Metric: It’s a function that satisfies the properties (non-negativity, symmetry, and triangle inequality) we might expect of a distance.

Metric Rules

• A metric space $(X, d)$ is a set $X$ with a metric $d$ defined on $X$.
• The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance.
• A metric may correspond to a metaphorical مجازي, rather than a physical notion of distance
  • Example 1: A set of 100-character Unicode strings can be equipped with the Hamming distance
  • Side Note: Hamming Distance measures the number of characters that need to be changed to get from one string to another.
  • Formula: Hamming distance d(str1, str2) = Σᵢ (str1[i] ≠ str2[i])

• Example 2: We can define a for any set X a trivial metric called Discrete Metric
  • Formula: Discrete distance d(x, y) = { 1 if x = y 0 if x = y }
• Example 3: We can define a for any set X a trivial metric called Discrete Metric
  • Formula: Discrete distance d(x, y) = { 1 if x = y 0 if x = y }

• In general, there are no algebraic operations defined on a metric space, only a distance function. Most of the spaces that arise in the analysis are vector, or linear, spaces, and the metrics on them are usually derived from a norm, which gives the “length” of a vector

Open Ball vs Neighborhood

In a topological space, the concept of an open ball is generalized to define neighborhoods around points. Unlike in metric spaces where an open ball is defined using a distance function, in a topological space, neighborhoods are defined based on the open sets of the space.

In a topological space, an open ball around a point x is defined as a set that contains a neighborhood of x. Formally, given a topological space (X, τ), an open ball around x is denoted as B(x) and is defined as:

B(x) = {y ∈ X | y is in a neighborhood of x}

In other words, the open ball B(x) includes all points that are in some neighborhood of x. The specific shape or size of the open ball depends on the structure of the topological space and the open sets defined within it.

It's important to note that the open ball in a topological space may not resemble the traditional concept of a ball in Euclidean space. The shape and size of the open ball are determined by the open sets defined in the given topological space.

In summary, in a topological space, an open ball around a point x is a set that contains a neighborhood of x, where a neighborhood is a set that is sufficiently "close" to x based on the open sets defined in the topological space.

Normed Vector Space

https://www.youtube.com/watch?v=6U4DaU3B_T0&ab_channel=JackNathan

• Neighborhoods, Continuity, Compactness, Connectedness, Homeomorphisms, Stretching

Vector Spaces

Intro to Vectors

[1] Euclidean Space (Classical / Plane Geometry)

• An orthogonal coordinate system refers to a system where the coordinate lines meet at right angles (90 degrees).

Curvilinear Coordinate System

• It’s a mathematical framework used to describe points using coordinates that are defined in relation to curved surfaces or curves.
• In contrast to the Cartesian coordinate system, which uses straight lines as reference axes (coordinates), curvilinear coordinate systems utilize curves or curved surfaces as reference lines or surfaces.
• The main difference is that the coordinate lines in linear coordinates are straight lines with constant scaling, while the coordinate lines in curvilinear coordinates are curvy lines with changing scalings.
• For each curvilinear coordinate system, we will have new coordinates:
  • These new coordinates shall be derived from a set of Cartesian coordinates by using a transformation that is a one-to-one map at each point.
  • This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back.
• https://www.youtube.com/watch?v=3diDc4jmEgk&ab_channel=NPTEL-NOCIITM
• This curvilinear framework involves different coordinate systems such as:
  • 1.b — Polar Coordinate System
  • 1.c — Cylindrical Coordinate System
  • 1.d — Spherical Coordinate System
  • 1.e — Parabolic Coordinate System
  • 1.f — Bipolar Coordinate System
  • 1.e — Elliptical Coordinate System
  • 1.g — Toroidal coordinates
• A curvilinear coordinate system is simpler to use than a Cartesian coordinate system for some applications.
• Ex: while one might describe the motion of a particle in a rectangular box using Cartesian coordinates, the motion in a sphere is easier with spherical coordinates.
• Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography رسم الخرائط, quantum mechanics, and relativity.

[1.g] Elliptic Coordinate System [2D] & Elliptic Cylindrical Coordinate System [3D]

Not to be confused with the Ecliptic coordinate system نظام إحداثيات الكسوف.

Elliptic Coordinate System

• We had to study what the ellipses and hyperbola are to digest this elliptic coordinate system.
• The elliptic coordinate system is another type of curvilinear coordinate system that extends the concept of coordinate systems to two dimensions.
• Similar to other curvilinear coordinate systems, the elliptic coordinate system provides an alternative way to describe points in the plane beyond the traditional Cartesian (x, y) coordinates.
• The elliptic coordinate system uses two coordinate variables, usually denoted as $(μ, ν)$, to specify the position of a point in the coordinate plane. with respect to certain reference curves or surfaces.
• The coordinate curves are orthogonal to each other, meaning they intersect at right angles.

• In Elliptic coordinate systems, the coordinate lines are confocal ellipses and hyperbolae.

[1.f] Bipolar Coordinate System [2D] & Bipolar Cylindrical coordinates [3D]

No need to study

Curve coordinate system: This is a coordinate system with directions (s, t) where s always follows the orientation of a defined arclength parameterized curve and t is orthogonal to s2. — It’s 2D curvilinear

[1.e] Log-Polar Coordinate System [2D]

represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.

[1.e] Barycentric Coordinate System

[1.e] Trilinear Coordinate System

Coordinate Systems that are Neither Linear nor Curvilinear:

• Homogeneous Coordinates
• Projective Coordinates
• Normalized Device Coordinates (NDC)
• Clip Coordinates
• Log-polar Coordinates
• Barycentric Coordinates
• Intrinsic Coordinates
• Geographical Coordinates (Latitude, Longitude, Altitude)
• polar stereographic
• Mercator projections.

parabolic coordinates, toroidal coordinates, bipolar coordinates

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular z-direction.

Homogeneous Coordinate System: This system is commonly used in computer graphics and computer vision. It represents points or transformations using homogeneous coordinates, which include an additional coordinate to handle translation and projective transformations.

[2] Spherical Space

• Spherical geometry is based on the surface of a sphere.
• Spherical geometry exhibits positive curvature and has properties that differ from Euclidean geometry, including the sum of angles in a triangle being greater than 180 degrees.
• A great circle divides the sphere into two equal halves.
• There are infinitely many great circles on the surface of a sphere.
• For example, the equator is a great circle on the Earth's surface.

• Plane (Euclidean) geometry: the basic concepts are points and (straight) lines.
• Spherical geometry: the basic concepts are point and great circle.
• In mathematics, the symbol commonly used to denote a spherical space is $S^n$, where n represents the dimension of the space.
• $S^0$: This represents a zero-dimensional spherical space, which consists of just two points (usually referred to as the "north pole" and "south pole").
• $S^1$: : This represents a one-dimensional spherical space, commonly known as a circle.
• $S^2$: : This represents a two-dimensional spherical space, which is often referred to as a sphere.
• $S^3$:: This represents a three-dimensional spherical space, known as a hypersphere or simply a sphere in four-dimensional space.
• $S^4$: This represents a four-dimensional spherical space, sometimes referred to as a 4-sphere.
• $S^3$:

Yes, there is something called spherical space. It is a geometric space that is topologically equivalent to the surface of a sphere. This means that it is a two-dimensional space that can be continuously deformed into the surface of a sphere without any tears or breaks.

Spherical space is often used in mathematics and physics to model the universe. In this context, the sphere represents the celestial sphere, which is the apparent surface of the sky as seen from Earth. The points on the sphere represent the positions of stars and galaxies.

Spherical space is also used in computer graphics to represent the surface of a sphere. This is done by dividing the sphere into a number of triangles and then assigning each triangle a color or texture.

Here are some of the properties of spherical space:

• The sum of the interior angles of a triangle in spherical space is greater than 180 degrees.
• There are no parallel lines in spherical space.
• The shortest distance between two points in spherical space is along a great circle.

Spherical space is a fascinating and important concept in mathematics and physics. It has a wide range of applications, and it continues to be studied and explored by mathematicians and physicists today.

In spherical space, points are typically represented using coordinates that correspond to the angles or directions from a reference point. The specific coordinate system used depends on the dimension of the spherical space. Here are some common coordinate systems for representing points in spherical spaces:

1. Spherical coordinates: In three-dimensional spherical space (�2S2), points are often represented using spherical coordinates. The coordinates consist of two angles: the azimuth angle (measured horizontally around a reference axis) and the polar angle (measured vertically from a reference plane). These coordinates are typically denoted as (�,�)(θ,ϕ), where �θ represents the azimuth angle and �ϕ represents the polar angle.
2. Hyperspherical coordinates: In higher-dimensional spherical spaces, such as �3S3 or ��Sn, more generalized coordinate systems are used. Hyperspherical coordinates extend the concept of spherical coordinates to higher dimensions. They involve a set of angles or parameters that describe the position of a point in the hypersphere.

It's important to note that the specific coordinate system and representation can vary depending on the context and mathematical formulation being used. The choice of coordinate system depends on the dimension of the spherical space and the mathematical convenience of the problem at hand.

Shortest Path?

• In spherical geometry, the shortest path between two points is along a great circle.
• When you want to find the shortest path between two points on the surface of a sphere, you can think of it as walking along the sphere's surface.
• IMPORTANT: The path that follows the arc نفس الاتجاه of the great circle connecting those two points is the shortest distance between them. مش اي دائرة كبيرة وخلاص، بل نتبع نفس اتجاه الدائرة الكبرى التي تصل بين هاتين النقتطين للحصول على أقصر مسافة
• This concept is analogous to how a straight line is the shortest distance between two points in Euclidean geometry. However, on a sphere, due to its curved nature, the shortest distance is achieved by following the curve of a great circle rather than a straight line.

Parallel Lines?

• No, parallel lines do not intersect in spherical geometry.
• Actually, spherical geometry does not have parallel lines in the traditional sense.
• However, in the context of spherical geometry, we can consider lines of latitude خط العرض as "parallel" since they do not intersect each other and maintain an equal distance from the equator.
• Similarly, lines of longitude can be considered "meridians خطوط الطول" and are also parallel to each other, but they converge at the poles.

[3] Affine Space

[3] Manifold

[3] Hyperbolic Space

• Hyperbolic geometry is characterized by negative curvature.
• The hyperbolic plane is a non-Euclidean geometry where the parallel postulate is violated. It has different rules for angles and distances, leading to unique geometric properties. The hyperbolic plane is useful in the study of certain curved surfaces and in non-Euclidean geometry.

[4] Elliptic Space

[5] Projective Space

[5] Minkowski Space

Affine Space, Baire Space, Banach Space, Base Space, Bergman Space, Besov Space, Borel Space, Calabi-Yau Space, Cellular Space, Chu Space, Dimension, Drinfeld's Symmetric Space, Eilenberg-Mac Lane Space, Euclidean Space, Fiber Space, Finsler Space, First-Countable Space, Fréchet Space, Function Space, G-Space, Green Space, Heisenberg Space, Hilbert Space, Inner Product Space, L2-Space, Lens Space, Liouville Space, Locally Finite Space, Loop Space, Mapping Space, Measure Space, Metric Space, Minkowski Space, Müntz Space, Non-Euclidean Geometry, Normed Space, Paracompact Space, Planar Space, Polish Space, Probability Space, Projective Space, Quotient Space, Riemann's Moduli Space, Sample Space, Standard Space, State Space, Stone Space, Symplectic Space, T2-Space, Teichmüller Space, Tensor Space, Topological Space, Topological Vector Space, Total Space, Vector Space

[5.a] Homogeneous (Projective) Coordinate System

The homogeneous coordinate system is a mathematical representation used in computer graphics, computer vision, and projective geometry. It extends the traditional Cartesian coordinate system to include an additional coordinate, known as the homogeneous coordinate. The homogeneous coordinate system allows for convenient representation and manipulation of points at infinity and transformations.

In the Cartesian coordinate system, a point is typically represented using three coordinates (x, y, z) in three-dimensional space or two coordinates (x, y) in two-dimensional space. However, the homogeneous coordinate system introduces a fourth coordinate (w) along with the traditional coordinates (x, y, z).

https://codinghero.ai/8-different-types-of-coordinate-systems-explained-to-kids/#4_Homogeneous_Coordinate_System

Projective Plane: The projective plane extends the Euclidean plane by adding "points at infinity." It allows for the representation of parallel lines intersecting at a point at infinity. The projective plane has applications in projective geometry and computer graphics.

Taxicob Space (Manhattan / L1 / Rectilinear Geometry)

The taxicab distance formula is based on the idea that the distance between two points is found by following a grid, rather than following a straight line. The formula is the sum of the absolute value of the difference between x values and the absolute value of the difference between y values

The distance between two points is found not by a straight line drawn between the points, but as if the points lie on a grid. This grid is similar to a grid of streets.

Manhattan geometry: This term refers to taxicab geometry's association with the layout of streets in Manhattan, where taxis often navigate a grid system of streets.

L1 geometry: The name L1 geometry comes from the fact that the distance metric used in taxicab geometry is sometimes denoted as L1 distance, where L1 represents the norm or metric used to calculate the distance.

Rectilinear geometry: Rectilinear geometry is another name for taxicab geometry, emphasizing the property that movement is restricted to horizontal and vertical directions along a grid, forming right angles.

While Euclidean space was the only geometry for thousands of years, non-Euclidean spaces have some useful applications. For example, taxicab geometry allows you to measure distance when you can only move vertically or horizontally; It’s applications include calculating distances or boundaries anywhere you can’t move “as the crow flies”, like by car in New York City.