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Non Euclidean Geometry
... by cutting the polygon into n–2 triangles. A spherical polygon with n sides can be cut in the same way into n–2 spherical triangles, each of which has angle sum more than 180°, and so the angle sum of a spherical n-gon is more than (n – 2)180°. Put another way, the angle sum of a spherical polygon ...
... by cutting the polygon into n–2 triangles. A spherical polygon with n sides can be cut in the same way into n–2 spherical triangles, each of which has angle sum more than 180°, and so the angle sum of a spherical n-gon is more than (n – 2)180°. Put another way, the angle sum of a spherical polygon ...
Free Topological Groups - Universidad Complutense de Madrid
... This is a good point to turn back to the problem of the existence of free topological groups. Let us consider the non-Abelian case first. It follows from Definition 1.1 that the topology of the group F (X ) (when the latter exists) is maximal in some sense. Here is the exact mathematical formulation ...
... This is a good point to turn back to the problem of the existence of free topological groups. Let us consider the non-Abelian case first. It follows from Definition 1.1 that the topology of the group F (X ) (when the latter exists) is maximal in some sense. Here is the exact mathematical formulation ...
Topology .
... (i) if K is a compact subset of X and y ∈ X a point outside K then y and K have disjoint neighborhoods, i.e. there is an open neighborhood W y of y and an open set Vy ⊃ K with Wy ∩ Vy = ∅. (ii) every compact subset of X is closed. (iii) any two disjoint compact subsets of X have disjoint open neighb ...
... (i) if K is a compact subset of X and y ∈ X a point outside K then y and K have disjoint neighborhoods, i.e. there is an open neighborhood W y of y and an open set Vy ⊃ K with Wy ∩ Vy = ∅. (ii) every compact subset of X is closed. (iii) any two disjoint compact subsets of X have disjoint open neighb ...
A Demonstration that Quotient Spaces of Locally Compact Hausdorff
... Theorem 4.8. Suppose f : X → Y is a continuous surjective function. If X is compact and Y is Hausdorff, then f is a quotient map. Proof. Let C be a closed subset of X. A closed subset of a compact space is compact, so C is compact. The continuous image of a compact set is compact, so f (C) is a comp ...
... Theorem 4.8. Suppose f : X → Y is a continuous surjective function. If X is compact and Y is Hausdorff, then f is a quotient map. Proof. Let C be a closed subset of X. A closed subset of a compact space is compact, so C is compact. The continuous image of a compact set is compact, so f (C) is a comp ...
3-manifold
![](https://commons.wikimedia.org/wiki/Special:FilePath/3-Manifold_3-Torus.png?width=300)
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.