Course Outline

Metric and Topological Spaces

Deﬁnition and examples. Limits and continuity. Open sets and neighbourhoods. Characterizing limits and continuity using neighbourhoods and open sets. [3]

Deﬁnition of a topology. Metric topologies. Further examples. Neighbourhoods, closed sets, convergence and continuity. Hausdorﬀ spaces. Homeomorphisms. Topological and non-topological properties. Completeness. Subspace, quotient and product topologies. [3]

Deﬁnition using open sets and integer-valued functions. Examples, including intervals. Components. The continuous image of a connected space is connected. Path-connectedness. Path-connected spaces are connected but not conversely. Connected open sets in Euclidean space are path-connected. [3]

Deﬁnition using open covers. Examples: ﬁnite sets and [0,1]. Closed subsets of compact spaces are compact. Compact subsets of a Hausdorﬀ space must be closed. The compact subsets of the real line. Continuous images of compact sets are compact. Quotient spaces. Continuous real-valued functions on a compact space are bounded and attain their bounds. The product of two compact spaces is compact. The compact subsets of Euclidean space. Sequential compactness. [3]

† W.A. Sutherland Introduction to metric and topological spaces. Clarendon 1975
D.J.H.Garling A Course in Mathematical Analysis (Vol 2). Cambridge University Press 2013
A.J. White Real analysis: an introduction. Addison-Wesley 1968 B. Mendelson
Introduction to Topology. Dover, 1990