Basic Topology & Algebra

Hi! Here we are going to learn the basics in Topology & algebra, Here are given all essential definitions that must be known to a topology & algebra learner Such as-Types of Relations,. Check out all them at a glance……

Concepts in topology & Algebra:

Sets:

a set is an important concept in topology & algebra,which means a collection of well distinct objects. And object of a set are called elements.

Subset:

The topological concept of subset  can be better illustrated by an example…..

If every element of set A is is an element of set B, then A is called a subset of B

Super set:

The concept of Superset  in topology & algebra can be better understand using above example:

If A is a subset of B i.e. A  ⊆ B;

But, B is a super set of A i.e. B ⊇ A

Equality of sets:

If every element of A contained in B and each element of B is contained in A, Then sets A and B are said to be equal. i. e. A=B.

Proper Subset:

A subset B of A is a proper subset iff, B ⊆ A also A≠B

Power Set:

The family of all subsets of a set is called the power set.

It must be remembered that,each power set contains 2n elements.

Universal Set:

If all sets under consideration are subsets of a fixed set then, that set is a Universal set.

Singleton Set:

In topology,A set consisting of only one element, is a singleton set.

Complement of a set A:

The set: X-A=A’

Symbolically;   A’= X-A ={x ε X: x ≠ A}

Some tips about complement of a set:

  • X’ = Φ i.e. Φ’ = X i.e. Complement of whole set is a null set, & Vice-versa.
  • A U A’ = X
  • A ∩ A’=Φ
  • A – B=A ∩ B’
  • B – A =B ∩ A’
  • (A’)’=A

Types of Relations:

Relations are the basic ties in algebra & topology. Now, let us know about the relations & their types,

Reflexive:

It can be defined as,for any x ε X ; the relation between x with itself is known as Reflexive. And can be given as: x R x ,where R is any relation.

Symmetric:

If for some x & y belonging to a same set,say R (set of real nos.)then, if the relation between x & y is equal to that of relation between y & x then the relation is said to be  symmetric,                                                                                                   x R y = y R x

Anti-Symmetric:

If for some x & y  belonging to a same set,say R (set of real nos.)then, if the relation between x & y is equal to that of relation between y & x which implies that x and y are equal. i.e.                                                                                                                       x R y = y R x ⇒ x=y

Transitive:

If for some x, y & z belonging to a same set,say R (set of real nos.)then, if x and y are related, and y and z are related, then we can say that, x and z are related.            x R y & y R z ⇒ x R z                                                                                                                       

Equivalence Relation:      

In topology,the equivalence relation is a relation which is reflexive, symmetric and transitive relation.

Equivalence Classes:

Equivalent Classes are classes on the basis of the equivalence relation(i.e. reflexive,symmetric and transitive).

Partition of Set:

If Br(⊆A) is a non empty set for each value of r in an index set ∧. Then a family of non empty sets {B: r ε ∧} is  a partition of A , if…                                                        1. A=Ur∈ΔBr
2.For any   Br,Bs∈{Br}r∈Δ      ⇒     either Br=Bs  or
Br ∩ Bs = Φ

Partially ordered:

A set which is reflexive, anti symmetric and transitive is a partially ordered set.

Above are the basic topological concepts….But there are still more definitions to know about…

Read all of them…