A STUDY ON RETRACTIONS AND FIXED POINT

Author (s) : DR.B.VENKATACHALAM, V.SARANYA, S.SARANYA

---

Abstract :
We now prove several classical results of topology that follow from our knowledge of the fundamental group of S¹.The fundamental group is the first and simplest homotopy  group. The fundamental group is a homotopy invariant-topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by π1 (X). We now prove several classical results of topology that follow from our knowledge of the fundamental group of S¹. The tychonoff  theorem asserts that the product of an arbitrary number of compact spaces is compact in the product topology. In lecture three we have proved this result for finitely many space, but unfortunately, the same method does not work for infinite products .The  purpose of this project is to give a proof to the Tychonoff  theorem  in several steps. We used Zorn lemma to the tychonoff theorem.The stone- Čech compactification of  X ,which we study now, is in some sence the maximal compactification of  X. It was constructed by M.Stone  and  E. Čech, independently, in 1937. It has a number of applications in modern analysis, but these lie outside the scope of this book.

Keywords : Homotopy invariant-topological, The stone- Čech compactification

Conference Date : 13/11/2020

Pages : 101

Paper ID : 20101