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# Topology

### Spring 2003

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Lecturer: Sigmundur Gudmundsson

Literature: James R. Munkres, Topology, Prentice Hall (2000). [The book is available on Internet]

The Euclidean norm defined on the real line R, in the plan R2 or in the general n-dimensional space Rn leads to the notion of an open subset in these spaces and continuous functions between them. These play an important role in analysis.

In this course we shall introduce the general concept of a topological space (X,T) consisting of a set X together with a collection T of so called open subsets of X satisfying certain rules. Important examples of topological spaces are the metric spaces (X,d) generalizing the above mentioned cases of the real line, the plane, the n-dimensional space.

A map f:X->Y between two topological spaces is said to be continuous if the inverse image of an open subset of Y is open in X. We study properties of different topological spaces and of continuous maps f:X->Y between them. It turns out that such maps will in many cases have properties similar to those in the well-known Euclidean cases.

Keywords: cardinals, topological spaces, continuous maps, Hausdorff spaces, convergence, connectedness, compactness, metric spaces, the uniform limit theorem, completeness, Ascoli's theorem.

Written exams:

VIPs: Bolzano (1781-1848), Cauchy (1789-1857), Weierstrass (1815-1897), Heine (1821-1881), Schröder (1841-1902), Ascoli (1843-1896), Cantor (1845-1918), Arzela (1847-1912), Hilbert (1862-1943), Minkowski (1864-1909), Hausdorff (1868-1942), Lindelöf (1870-1946), Borel (1871-1956), Lebesgue (1875-1941), Bernstein (1878-1956), Banach (1892-1945), Cech (1893-1960), Urysohn (1898-1924), Tietze (1880-1964), Stone (1903-1989), Gödel (1906-1978), Tychonoff (1906-1993), Cohen (1934- ).