Ways to “compress” a topology!

# Category: Pure Mathematics

## Quick Revision: Topological Spaces (Part 1.1)

The definition of topological spaces and some interesting examples and counterexamples of topological spaces!

## Quick Revision: Topology (Part 0)

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

## Quick Revision: Topology

An intuitive introduction to topology/quickly revise the paper after a semester of coursework.

## Animation: X – (A ∪ B) = (X – A) ∩ (X – B)

Proof for X – (A U B) = (X – A) ∩ (X – B) using Venn Diagrams / Animation.

## 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…

## Puzzle: Finding a Bounded Continuous Function

Find a continuous, real-valued with its range contained in (0, 1).