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…….


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


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.


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,
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.


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.


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….