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... that f −1 is also continuous. We also say that two spaces are homeomorphic if such a map exists. If two topological spaces are homeomorphic, they are topologically equivalent — using the techniques of topology, there is no way of distinguishing one space from the other. An autohomeomorphism (also kn ...
... that f −1 is also continuous. We also say that two spaces are homeomorphic if such a map exists. If two topological spaces are homeomorphic, they are topologically equivalent — using the techniques of topology, there is no way of distinguishing one space from the other. An autohomeomorphism (also kn ...
k h b c b a q c p e a d r e m d f g n p r l m k g l q h n f
... homotopic to IdX and IdY , respectively. S 2 minus n points is homeomorphic to R2 minus n − 1 points, which deformation retracts onto a wedge of n − 1 copies of S 1 . The fundamental group of the Möbius band is Z, while that of the projective plane is Z/2. The two spaces are therefore not homotopy ...
... homotopic to IdX and IdY , respectively. S 2 minus n points is homeomorphic to R2 minus n − 1 points, which deformation retracts onto a wedge of n − 1 copies of S 1 . The fundamental group of the Möbius band is Z, while that of the projective plane is Z/2. The two spaces are therefore not homotopy ...
Topology Ph.D. Qualifying Exam ffrey Martin Geo Mao-Pei Tsui
... 11. Prove that a topological space X is disconnected iff there is a non-constant continuous map from X to {0, 1} where the latter space is given the discrete topology, that is, every subset is open. 12. A topological space X is T 1 if given any two distinct points each has a neighborhood that is dis ...
... 11. Prove that a topological space X is disconnected iff there is a non-constant continuous map from X to {0, 1} where the latter space is given the discrete topology, that is, every subset is open. 12. A topological space X is T 1 if given any two distinct points each has a neighborhood that is dis ...
Theorem 1. (Exterior Angle Inequality) The measure of an exterior
... Proof: By Lemma 2, the angle sum of 4ABC ≤ 180◦ and the angle sum of 4ACD ≤ 180◦ . If both of these inequalities were equalities we would have m∠1 + m∠2 + m∠B + m∠D + m∠3 + m∠4 = 360, in which case the angle sum of 4ABD = m∠1 + m∠2 + m∠B + m∠D = 360 − m∠3 − m∠4 = 180◦ , contradicting our hypothesis. ...
... Proof: By Lemma 2, the angle sum of 4ABC ≤ 180◦ and the angle sum of 4ACD ≤ 180◦ . If both of these inequalities were equalities we would have m∠1 + m∠2 + m∠B + m∠D + m∠3 + m∠4 = 360, in which case the angle sum of 4ABD = m∠1 + m∠2 + m∠B + m∠D = 360 − m∠3 − m∠4 = 180◦ , contradicting our hypothesis. ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.