Metric Space These are actually based on the lectures delivered by Prof. Muhammad Shoaib (HoD, Department of Mathematics , University of Okara). These notes are very helpful to prepare a section of paper mostly called Topology in MSc. These are also helpful in BSC or Bs(Hons) Classes CONTENTS OR SUMMARY: Metric Spaces and examples Pseudometric and example Distance between sets Theorem: Let ( X , d ) ( X , d ) be a metric space. Then for any x , y ∈ X x , y ∈ X , | d ( x , A ) − d ( y , A ) | ≤ d ( x , y ) . | d ( x , A ) − d ( y , A ) | ≤ d ( x , y ) . Diameter of a set Bounded Set Theorem: The union of two bounded set is bounded. Open Ball, closed ball, sphere and examples Open Set Theorem: An open ball in metric space X is open. Limit point of a set Closed Set Theorem: A subset A of a metric space is closed if and only if its complement A c A c is open. Theorem: A closed ball is a closed set. Theorem: Let ( X,d ) be a metric space and A ⊂ X A ⊂ X . If x ∈ X x ∈ X is a lim
