Basic Topology & Algebra 2

Before reading this article, hope you have gone through the previous section of the definitions in topology …..Here are some more definitions such as mapping, cardinal & ordinal nos, equipotent etc, that would provide the ease in understanding the subject…….

 Domain:

The set consisting of all the first elements.(elements that are not mapped with any function.)

Co-domain:

Set consisting of the elements that are transformed under some mapping.

Functions in topology:

Function in topology, is nothing but a mapping from domain to co-domain,which transforms the elements of domain into the elements of co-domain.

Range:

the number of elements in the set is termed as the range of the set.
e.g.:consider a set of natural no.
i.e.:N= {1,2,3,4,5,6,7,8,9
then,the range of the above set is 9.

Topological mappings:

onto mapping:  (Surjection)

If  f: x → y is a mapping,
f is an onto mapping if f(x)=Y i.e.every element in Y must have a pre-image in X.

 into mapping: (injection)

If  f: x → y is a mapping,
f is an into mapping if f(x)=Y there exist at least one element in Y which has no pre-image in X.

 one-one mapping: (Bijective)

If  f: x1→ x2 is a mapping,
f(x1)=f(x2)
⇒x1=x2
In other words, one-one function is both onto & into.

Cardinally equivalent:

Set A would be Cardinally equivalent to B if there exist one one mapping from A to B is denoted by : A∼B.

Denumerable:

A set is denumerable if A∼N i.e. there exist one-one mapping.
OR A set equipotent to the set of natural numbers.

At most countable:

A set in topology, at most countable if it is either finite or countable.

Uncountable:

A topological set A would be uncountable if it is neither finite nor countable.

Cardinal numbers:

In topology,a Cardinal Number is,a measure of number of points in a set.

Some properties of Cardinal numbers:

  • Basic property of Cardinal no. is that, if A∼B
    ⇒ Range of A =Range of B
  • ‘Transfinite’ Cardinal number is the cardinal number of an infinite set
  • Set of all rational no. is Denumerable.
  • The union of denumerable no. of denumerable set is a denumerable set.
  • Every infinite set contains a denumerable subset.
  • Every infinite set is equipotent to the set of proper subset of itself.
  • The set of real numbers is uncountable.
  •  Real & Complex nos. are equipotent.
  • Algebraic numbers are denumerable.

above all is about basics, Now, we turn towards the topological terms…

Read them….