Do you know topology? 2 "Analysis Situs"

Probability Level 3

Let the sphere S 2 = { ( x , y , z ) ∈ R 3 ; x 2 + y 2 + z 2 = 1 } \mathbb{S}^2 = \{(x,y,z) \in \mathbb{R}^3 ; x^2 + y^2 + z^2 =1 \} and the torus T 2 = { ( x , y , z ) ∈ R 3 ; ( x 2 + y 2 − a ) 2 + z 2 = r 2 and a > r > 0 } \mathbb{T}^2 = \{(x,y,z) \in \mathbb{R}^3 ; (\sqrt{x^2 + y^2} - a)^2 + z^2 = r^2 \text{and } a > r >0\} .

Which statements below are true?

a) There exists an homeomorphism from S 2 \mathbb{S}^2 to T 2 \mathbb{T}^2

b) There exists an homeomorphism from S 2 \mathbb{S}^2 to R 2 \mathbb{R}^2

Details and assumptions :

The topology used in each space is the euclidean topology

Both (a) and (b) (b) Neither (a) nor (b) (a)

Guillermo Templado by Guillermo Templado , 46, Spain

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1 solution

a) S 2 is simply connected and T 2 isn’t simply connected \mathbb{S}^2 \text{ is simply connected and } \mathbb{T}^2 \text{ isn't simply connected} and simply connectedness is an unvariant under homeomorphisms(bijective continuous function)... (we can also say that they have distinct fundamental groups or Euler's Characteristic are distinct (Vertex -Edges + Faces = χ \chi ; χ ( S 2 ) = 2 ; χ ( T 2 ) = 0 \chi(\mathbb{S}^2) = 2 ; \chi(\mathbb{T}^2) = 0 and group fundamental an Euler characteristic are topological invariant...) EULER CHARACTERISTIC

b) S 2 is a compact set (with euclidean topology it is closed and bounded) and R 2 isn’t it \mathbb{S}^2 \text{ is a compact set (with euclidean topology it is closed and bounded) and } \mathbb{R}^2 \text{ isn't it} and compactness is an unvariant under homeomorphisms

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