When I was a student, I studied the subject using a very old book "Topological groups", written by L. S. Pontryagin. It was in Russian, but Google search shows that there is a translation in English.
I just got this through online. I think this book will help me a lot. Thank you very much sir.
The most important topological groups are Lie Groups. I would start with them and then generalize. There is a text book which introduces Lie Group using matrixes as examples:-- "Lie Groups: A problem-Oriented Introduction via Matrix Groups" by Harriet Pollatsek. Note that you will see Lie Algebras also discussed with Lie Groups, but this association does not generalize to topological groups.
- Alexander Arhangel’skii
- Mikhail Tkachenko
Topological Groups and Related Structures Book Topological Groups and Related Structures
I would recommend a very recent, carefully written informative and friendly book
"Topological Groups and the Pontryagin-van Kampen Duality: An Introduction" by Lydia Außenhofer, Dikran Dikranjan, Anna Giordano Bruno
Book Topological Groups and the Pontryagin-van Kampen Duality: An...
https://link.springer.com/content/pdf/bbm%3A978-94-91216-35-0%2F1.pdf
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