Non-Euclidean Geometry - Department of Mathematics | Illinois
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 ...
A Demonstration that Quotient Spaces of Locally Compact Hausdorff
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 ...
Basic Algebraic Geometry
Basic Algebraic Geometry

Consonance and Cantor set-selectors
Consonance and Cantor set-selectors

Reflections on manifolds
Reflections on manifolds

Available online through www.ijma.info ISSN 2229 – 5046
Available online through www.ijma.info ISSN 2229 – 5046

Geometry and Topology, Lecture 4 The fundamental group and
Geometry and Topology, Lecture 4 The fundamental group and

Lecture 3
Lecture 3

Cardinal properties of Hattori spaces on the real lines and their superextensions
Cardinal properties of Hattori spaces on the real lines and their superextensions

An introduction to differential topology
An introduction to differential topology

On Pre-Λ-Sets and Pre-V-sets
On Pre-Λ-Sets and Pre-V-sets

Axiomatic Approach to Homology Theory Author(s)
Axiomatic Approach to Homology Theory Author(s)

Countable dense homogeneous filters and the Menger covering
Countable dense homogeneous filters and the Menger covering

ON METRIZABLE ENVELOPING SEMIGROUPS 1. Introduction A
ON METRIZABLE ENVELOPING SEMIGROUPS 1. Introduction A

Contra-e-Continuous Functions 1 Introduction
Contra-e-Continuous Functions 1 Introduction

Teacher Talk-Standards behind Reasoning
Teacher Talk-Standards behind Reasoning

paracompactness with respect to anideal
paracompactness with respect to anideal

Adobe PDF
Adobe PDF

Configurations of points - University of Edinburgh
Configurations of points - University of Edinburgh

THE PRODUCT TOPOLOGY Contents 1. The Product Topology 1 2
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 ...
GEOMETRIC REPRESENTATION THEORY OF THE HILBERT SCHEMES PART I
GEOMETRIC REPRESENTATION THEORY OF THE HILBERT SCHEMES PART I

Lecture 11 COVERING SPACES A covering space
Lecture 11 COVERING SPACES A covering space

THE REGULAR OPEN-OPEN TOPOLOGY FOR FUNCTION
THE REGULAR OPEN-OPEN TOPOLOGY FOR FUNCTION

examples of non-polygonal limit shapes in iid first
examples of non-polygonal limit shapes in iid first

On α-normal and β-normal spaces
On α-normal and β-normal spaces

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.