![Non-Euclidean Geometry - Department of Mathematics | Illinois](http://s1.studyres.com/store/data/008399890_1-eaad312e3b28508d753d230716acf40b-300x300.png)
Non-Euclidean Geometry - Department of Mathematics | Illinois
... ◦ 3: Given a point and a distance a circle can be drawn with the point as center and the distance as radius ◦ 4: All right angles are equal ◦ 5: Given a point p and a line l, there is exactly one line through p that is parallel to l ...
... ◦ 3: Given a point and a distance a circle can be drawn with the point as center and the distance as radius ◦ 4: All right angles are equal ◦ 5: Given a point p and a line l, there is exactly one line through p that is parallel to l ...
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 ...
THE PRODUCT TOPOLOGY Contents 1. The Product Topology 1 2
... Definition 4.2. A topological space (X, τ ) is said to be second countable if τ has a countable basis. Proposition 4.3. Let (X, τ ) be a second countable T4 space, then X metrizable. Proof. Since Hilbert’s cube I ∞ is metrizable it suffices to show that X can be embedded in it. By the Embedding Lemm ...
... Definition 4.2. A topological space (X, τ ) is said to be second countable if τ has a countable basis. Proposition 4.3. Let (X, τ ) be a second countable T4 space, then X metrizable. Proof. Since Hilbert’s cube I ∞ is metrizable it suffices to show that X can be embedded in it. By the Embedding Lemm ...
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.