Metric Spaces

• Category: Science
• ISBN: N/A
• Views: 58
• Posted on: 14/09/2021 20:07

Definition of distance and metric spaces. Examples of Euclidean metric space, sequence spaces and function spaces. Open spheres. Open sets and their properties. Interior of a set and its properties. Closed set and its properties. Closure of a set and its properties. Boundary of a set. Dense set. (2 questions)

Base and axioms of countability. Separable space. Subspace. Product space. Sequences and subsequences and their convergence in a metric space. Cauchy sequence.Complete metric space and its examples.Cantor ’s theorem. Baire’s category theorem. (1 questions)

Definition and properties of a compact set. Compact sets in R. Finite intersection property. Bolzano-Weierstrass property and its equivalence. Sequential compactness and total boundedness. (2 questions)

Definition of continuous mapping in a metric space. Limit of a function in a metric space. Relationship between continuity and limit. Continuity in terms of open sets and closed sets. Continuity of constant and identity mappings.

Continuity and compactness. Connectedness in a metric space. Interval as a connected subspace.