Definition and examples. Continuity. The completion theorem, the Baire catedory theorem, the Banach fixed point rheorem, Compactness. Compactness in metric spaces. sequential compactness. Lebesgue number and total boudedness. Local compactness, the Aazela-Ascoli Theorem, the one-point compuctification. Connectedness. Path Connectedness, connected components. Tychonoff's Theorem. Basis, Uryssohn's Lemma and Teitze's Theorem. Compact Hausdorff spaces. Additional topics may include. The Stene-Weierstrass Theorem, Lindeloff's Theorem, compactifiations.

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