a note on strongly lindel¨of spaces
a note on strongly lindel¨of spaces

Metric Spaces, Topological Spaces, and Compactness
Metric Spaces, Topological Spaces, and Compactness

Chapter 2
Chapter 2

LOCAL HOMEOMORPHISMS VIA ULTRAFILTER
LOCAL HOMEOMORPHISMS VIA ULTRAFILTER

Simplicial Complexes
Simplicial Complexes

Detecting Hilbert manifolds among isometrically homogeneous
Detecting Hilbert manifolds among isometrically homogeneous

COMPACT LIE GROUPS Contents 1. Smooth Manifolds and Maps 1
COMPACT LIE GROUPS Contents 1. Smooth Manifolds and Maps 1

A New Type of Weak Continuity 1 Introduction
A New Type of Weak Continuity 1 Introduction

... Proof : Let {Vi |i ∈ Λ} be any cover of f (K) by open sets of (Y, σ). For each x ∈ X, there exists α(x) ∈ Λ such that f (x) ∈ Vα(x) . Since f is weakly sgα-continuous, there exists U (x) ∈ sgαO(X, x) such that f (U (x)) ⊂ sgαCl(Vα(x) ). The family {U (x)|x ∈ A} is a cover of A by sgα-open sets of X. ...
Math F651: Take Home Midterm Solutions March 10, 2017 1. A
Math F651: Take Home Midterm Solutions March 10, 2017 1. A

open cover
open cover

On Preclosed Sets and Their Generalizations
On Preclosed Sets and Their Generalizations

tychonoff`s theorem - American Mathematical Society
tychonoff`s theorem - American Mathematical Society

opensetsXX V1 andXXV2inXX Ywithw1EXX Vtandw2EXXV2. {x
opensetsXX V1 andXXV2inXX Ywithw1EXX Vtandw2EXXV2. {x

... Conversely, if X X Y is Hausdorff then so are the subspaces X X f y }, {x} XYandhencesoareXandY. Thus spaces like S' X S' X ... X S' are Hausdorff. Although subspaces of Hausdorff spaces are Hausdorff and products of Iiausdorff spaces are Hausdorff it is not true in general that a quotient space of ...
2. Permutation groups Throughout this section, assume that G is a
2. Permutation groups Throughout this section, assume that G is a

Chapter 7 Partitions of Unity, Orientability, Covering Maps ~
Chapter 7 Partitions of Unity, Orientability, Covering Maps ~

For the Oral Candidacy examination, the student is examined in
For the Oral Candidacy examination, the student is examined in

LOCAL HOMEOMORPHISMS VIA ULTRAFILTER CONVERGENCE
LOCAL HOMEOMORPHISMS VIA ULTRAFILTER CONVERGENCE

Notes on Stratified spaces.
Notes on Stratified spaces.

Extensions of totally bounded pseudometrics
Extensions of totally bounded pseudometrics

Part II - Cornell Math
Part II - Cornell Math

Examples and some basic properties of groups 1. Definition (Group
Examples and some basic properties of groups 1. Definition (Group

and x ∈ U y ∈ V ˆ N = N∪{∞} (d) Let a, b:ˆN
and x ∈ U y ∈ V ˆ N = N∪{∞} (d) Let a, b:ˆN

Problem set 3 - Math Berkeley
Problem set 3 - Math Berkeley

Topology Proceedings 7 (1982) pp. 293
Topology Proceedings 7 (1982) pp. 293

DEFINITIONS AND EXAMPLES FROM POINT SET TOPOLOGY A
DEFINITIONS AND EXAMPLES FROM POINT SET TOPOLOGY A

Fundamental group

In the mathematics of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group.Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. The abelianization of the fundamental group can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.Henri Poincaré defined the fundamental group in 1895 in his paper ""Analysis situs"". The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.