A series to quickly revise Topology – Dedicated to my students.

# Tag: Real Analysis

## Analysis/Topology: Bounded and Totally Bounded Sets

Definitions Bounded set: Let <M,ρ> be a metric space. We say that the subset A of M is bounded if there exists a positive number L such that ρ(x,y)≤L (x,y∈A) (Wikipedia) Diameter of a set: If A is bounded, we define diameter of A (denoted by diam A) as diam A= ρ(x,y). If A is not…

## Analysis/Topology: Connected Sets in Metric Spaces

A short discussion on Connected Sets in Metric Spaces with definition, examples, counterexamples, properties and quizzes.

## Topology/Analysis: Homeomorphism

Definition If is one-one and onto (1-1 correspondence/bijection) continuous is continuous Then we call a homeomorphism between two metric spaces . The metric space are said to be homeomorphic. Properties and Theorems If f is a homeomorphism between metric spaces , The set G⊂ is open if and only if the image f(G)⊂ is open. The…

## Video Lecture: Examples of Metric Spaces (Lecture 2)

Here’s a video discussing a few simple examples of metric spaces.

## Video Lecture: Definition of Metric Spaces (Lecture 1)

Here’s a video discussing the definition of metric spaces, and the inspiration behind it. Rarely

## Examples and Counterexamples: Real Line with Discrete Metric/Topology

Examples and Counterexamples: Real Line with Discrete Metric/Topology. Did you try the challenges at the end? Leave your thoughts in the comments below!

## CSIR-NET: Analysis/Topology Quiz 3

A simple question to check if you have understood the definition of metric spaces: how well do you know this definition?

## CSIR-NET: Analysis/Topology Quiz 2

Limit points of sequences in (−1,1) – CSIR-NET June 2016, Part B Question.

## CSIR-NET: Analysis/Topology Quiz 1

CSIR-NET December 2015, Part B Question. How should I analyze this question when I am preparing for CSIR-NET? Mark all mathematical terms here. Read their definitions. Try construct examples for each terms, try apply it in options. This question is interesting because it talks about a lot of mathematical terms, but the keyword is nowhere…