Planes and Spheres as Topological Manifolds. Stereographic Projection

Planes and Spheres as Topological Manifolds. Stereographic Projection The goal of this article is to show some examples of topological manifolds: planes and spheres in Euclidean space. In doing it, the article introduces the stereographic projection [25].


Preliminaries
Let us observe that ∅ is ∅-valued and ∅ is onto.Next we state three propositions: (1) For every function f and for every set Y holds dom(Y f ) = f −1 (Y ).
(2) For every function f and for all sets ) Let S, T be topological structures and f be a function from S into T .If f is homeomorphism, then f −1 is homeomorphism.Let S, T be topological structures.Let us note that the predicate S and T are homeomorphic is symmetric.
For simplicity, we use the following convention: T 1 , T 2 , T 3 denote topological spaces, A 1 denotes a subset of T Let g be a function from (5) For every function f from T 1 into T 2 such that f is homeomorphism holds f −1 (A 2 ) and A 2 are homeomorphic.(9) If T 1 is second-countable and T 1 and T 2 are homeomorphic, then T 2 is second-countable.
In the sequel n, k are natural numbers and M , N are non empty topological spaces.
The following propositions are true: (13) Let x 1 , x 2 be finite sequences of elements of R and i be an element of N.
(14) For all finite sequences x 1 , x 2 , y 1 , y 2 of elements of R such that len x 1 = len x 2 and len y 1 = len (15) For all finite sequences x 1 , x 2 , y 1 , y 2 of elements of R such that len x 1 = len x 2 and len y 1 = len In the sequel p, q, p 1 are points of E n T and r is a real number.One can prove the following propositions: (20) For every linear combination L 2 of R Seg n R and for every linear combination Brought  T .Suppose A 4 = A 5 .Then A 4 is linearly independent if and only if A 5 is linearly independent.(22) For every subset V of E n T such that V = RN-Base n there exists a linear combination l of V such that p = l.
(23) RN-Base n is a basis of E n T .( 24) Let V be a subset of E n T .Then V ∈ the topology of E n T if and only if for every p such that p ∈ V there exists r such that r > 0 and Ball(p, r) ⊆ V.
Let n be a natural number and let p be a point of E n T .The functor InnerProduct p yields a function from E n T into R 1 and is defined by: (Def. 1) For every point q of E n T holds (InnerProduct p)(q) = |(p, q)|.Let us consider n, p.Note that InnerProduct p is continuous.

Planes
Let us consider n and let us consider p, q.The functor Plane(p, q) yielding a subset of E n T is defined as follows: (Def.2) Plane(p, q) = {y; y ranges over points of E n T : |(p, y − q)| = 0}.The following propositions are true: Let us consider n and let us consider p, q.The functor TPlane(p, q) yields a non empty subspace of E n T and is defined by: (Def.3) TPlane(p, q) = E n T Plane(p, q).The following three propositions are true: (28) The base finite sequence of n + 1 and n + 1 = (0 T and TPlane(p, q) are homeomorphic.
(30) For all points p, q of E n+1 T such that p = 0 E n+1 T holds TPlane(p, q) is n-manifold.

Spheres
Let us consider n.The functor S n yields a topological space and is defined by: (Def.4) S n = TopUnitCircle(n + 1).
Let us consider n.Note that S n is non empty.Let us consider n, p and let S be a subspace of E n T .Let us assume that p ∈ Sphere((0 E n T ), 1).The functor σ S,p yielding a function from S into TPlane(p, 0 E n T ) is defined as follows: (Def.5) For every q such that q ∈ S holds (σ S,p )(q) = 1 1−|(q,p)| • (q − |(q, p)| • p).Next we state the proposition (31) For every subspace S of E n T such that Ω S = Sphere((0 E n T ), 1) \ {p} and p ∈ Sphere((0 E n T ), 1) holds σ S,p is homeomorphism.Let us consider n.One can verify the following observations: * S n is second-countable, * S n is n-locally Euclidean, and * S n is n-manifold.

( 6 )
If A 1 and A 2 are homeomorphic, then A 2 and A 1 are homeomorphic.(7) If A 1 and A 2 are homeomorphic, then A 1 is empty iff A 2 is empty.(8) If A 1 and A 2 are homeomorphic and A 2 and A 3 are homeomorphic, then A 1 and A 3 are homeomorphic.

( 10 )
If M is Hausdorff and M and N are homeomorphic, then N is Hausdorff.(11) If M is n-locally Euclidean and M and N are homeomorphic, then N is n-locally Euclidean.(12) If M is n-manifold and M and N are homeomorphic, then N is nmanifold.

( 21 )
Let A 4 be a subset of R Seg n R and A 5 be a subset of E n 1, A 2 denotes a subset of T 2 , and A 3 denotes a subset of T 3 .Let f be a function from T 1 into T 2 .Suppose f is homeomorphism.