Recent content by Wendel

The theorem: Let ##X##, ##Y## be sets. If there exist injections ##X \to Y## and ##Y \to X##, then ##X## and ##Y## are equivalent sets. Proof: Let ##f : X \rightarrow Y## and ##g : Y \rightarrow X## be injections. Each point ##x \in g(Y)⊆X## has a unique preimage ##y\in Y## under g; no ##x \in...

I think I understand now. Thank you fresh and Orodruin.

Or rather take the ball $$B = \{(x,y,z) \in ℝ^3 : x^2 +y^2 +z^2 ≤R^2 \} ⊆ ℝ^3$$ with R>0. Now choose three smaller numbers ##0 ≤ r_1 ≤ r_2 ≤ r_3 < R## and consider the spheres $$S^2_i = \{(x,y,z) \in ℝ^3 : x^2 + y^2 + z^2 = r^2_i\}, i \in \{1,2,3\}$$ Now Identify the boundary of ##B## with its...

Perhaps you could have A, B, C concentric in ℝ³ , describe another 2-sphere D outside of C, then identify all points in D with the point which is the center of sphere A?

Is it possible to have a topological space in which three 2-spheres A, B, C are such that B is in some sense nested inside A, C is nested inside B, but A is again nested in C. What about for three 2-tori in a similar manner?

Homework Statement Suppose we wish to construct a torus {(x,y,z)∈ℝ³:(R-√(x²+y²))²+z²=r²} whose axis of symmetry is the z-axis and the distance from the center of the tube to the center of the torus is R. Let r be the radius of the tube. Homework Equations The large circle in the xy-plane...

Thank you, it makes more sense now. However, I can't help but wonder, given a pair of points x,y∈X where X is a finite topological space, how to determine what other points in X must lie "between" x and y in order for it to be a path from x to y. Generally, the path wouldn't be surjective. I'm...

Homework Statement I'm trying to understand the intuition behind path-connectedness and simple-connectedness in finite topological spaces. Is there a general methodology or algorithm for finding out whether a given finite topological space is path-connected? Homework Equations how can I...

Hello, I'm a student interested in topology studying independently at the moment.