Tychonoff Theorem for Product Space

Dear Students
  • We are going to prove very important theorem called Tychonoff theorem which deal arbitrary product of compact spaces.
  • As we already discussed that product of a countably infinite family of topological spaces. We now proceed to define the product of any family of topological spaces by replacing the set {1, 2,… n,} by an arbitrary index set I. The central result will be the general Tychonoff Theorem.
  • This theorem help to construct the Stone-Cech compactification of a completely regular space and also in proving the general version of Ascoli’s theorem.
  • Download compete Lecture Notes. Tychonoff Theorem for Product Space

Compactness of Topological Space

Dear Students,

Now we are going study third “C” for the topological space as compactness properties. The compactness properties of the space is not as intuitive as continuity or connectedness. Compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

In this lecture we will learn Compactness of Topological Space.

Compactness of Topological Space


Connectedness of Topological Space

As simple as the idea of connectedness seems, it has deep applications for topology and its
applications. In calculus connectedness is an important tool to understand the IMVT in topological prospective. It is used to find the difference between two or more than two topological spaces. Further, many problems in geographic information systems, population modeling, and motion planning in robotics are possible to solve by connectedness.

Connectedness of Topological Spaces

Metric Topology

One of the most common and useful types of topological space is the so-called metric space. Metric spaces are topological spaces that result from having a means for measuring distance between points in the underlying set. This notion of measuring distance goes beyond stretching out a measuring tape to see how far apart two objects are.

  • We can measure the distance between two functions by considering the area bounded between their graphs.

  • We can measure the distance between two words by considering how many letter changes take us from one to the other.

  • The ability to measure and compare distances between elements of a set is often crucial, and it provides more structure than a general topological space possesses.

Metric Topology

Continuity of Topological Space

Dear Students,

As we have been defined various concepts related to topological space and their subsets. Now we would like to discus the functions sending one topological space to another. The concepts of continuity of functions is one of the most important in topology. In this post, we shall formulate a definition of continuity and study their various properties.

The Continuous Function on Topological Space



Topological Spaces and their Examples

Dear Friends,

Here you could learn about Topological Spaces and its interesting examples.  Apart of this you will learn  generation of topology from basis, new topological spaces from old, closed set, closure set, interior set  etc.

Topological Spaces and their Examples


Basis for Topology

We are familiar with the notion of a basis for a finite dimensional vector space. It is a minimal collection of vectors that spans the vector space. If we know a basis, we can always recover the vector space. A basis for a topology is in a quite similar fashion a family of open subsets that `span’ the topology. If we know a basis, then we can find the topology for given set. There is no good notion of minimality for a basis here, so there is no such requirement, but it is often convenient to have a basis with as few elements as possible.

Download the lecture notes form following link.

2 Basis for Topology LN




Topological Spaces and their Examples

In this lecture notes we are going to introduce the topological space which is primary object of study in field of topology. In first section, we give definition of topological space which is based on some axioms. In second section, we give interesting and interdisciplinary examples of topological spaces which help to understand basic theory of topology. You can download lecture note by clicking below.

1 Topological Space and their Examples