I-Notions of General Topology
Topology on a set. Comparison of the topologies on a set. Topological spaces. Topological subspaces. Neighborhoods. Neighborhood bases. Bases of a topology. Metrizable topological spaces. First and second countable spaces. Separable spaces. Continuity. Homeomorphisms of topological spaces. Product topology. Nets. Compact spaces. Tychonoff’s Theorem. Locally compact spaces. Alexandroff compactification. T1, Hausdorff, regular and normal spaces. Tietze extension theorem. Urysohn’s Lemma.
II- Topological Vector Spaces and Applications
Vector topology on a vector space. Topological vector spaces (t.v.s). Invariance of a vector topology under translation and scalar multiplication. Separation property of a t.v.s. Continuous linear mappings. Finite dimensional t.v.s. Metrizable t.v.s Locally convex spaces. F-spaces. Fréchet spaces. Bounded linear mappings. Seminormes. Minkowski’s functional. Locally convex vector topology induced by a separable family of semi-norms. Convergence in a locally convex space. Continuous linear forms on a locally convex space. Hahn-Banach theorem in a locally convex space and its consequences. The Fréchet spaces C(Ω) and C(Ω), where Ω is open in Rn . The closed subspaces DK(Ω) of C(Ω), where K is a compact set in (Ω) Baire’s theorem. Banach-Steinhaus theorem. Open mapping theorem. Closed graph theorem. Weak topology. Weak * topology. Banach-Alaoglu theorem. Reflexive spaces