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# Topology Lecture Note

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SKU: 25022021MATH0012 Categories: , ,

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Name of Notes : – Topology Lecture Note

Introduction

In mathematics, topology (from the Greek words τόπος, ‘place, location’, and λόγος, ‘study’) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homologies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.

The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler’s Seven Bridges of Königsberg problem and polyhedron formula are arguably the field’s first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.

Modules / Lectures

1 Topological Spaces

• 1.1 Basic Concepts
• 1.2 The Metric Topology
• 1.3 Interior Points, Limit Points, Boundary Points, Closure of a Set
• 1.4 Hausdorﬀ Topological Spaces .
• 1.5 Continuous Functions .

2 Product and Quotient Spaces

• 2.1 Product Space .
• 2.2 The Box Topology .
• 2.3 Quotient (Identiﬁcation) Spaces

3 Connected Topological Spaces

• 3.1 Connected Spaces
• 3.2 Connected Subsets of the Real Line
• 3.3 Some Properties of Connected Spaces .
• 3.4 Connected Components .

4 Compact Topological Spaces

• 4.1 Compact Spaces and Related Results
• 4.2 Local Compactness
• 4.3 One Point Compactiﬁcation of a Topological Space (X,J )
• 4.4 Tychonoﬀ Theorem for Product Spaces

5 Count ability and Separation Axioms

• 5.1 First and Second Countable Topological Spaces
• 5.2 Properties of First Countable Topological Spaces
• 5.3 Regular and Normal Topological Spaces . . .
• 5.4 Urysohn Lemma .
• 5.5 Tietze Extension Theorem .
• 5.6 Baire Category Theorem
• 5.7 Urysohn Metrization Theorem .