Mathematics, Faculty of Science

Sigmundur Gudmundsson

Topology

Spring 2015

 

Lecturer: Sigmundur Gudmundsson

Literature:
James R. Munkres, Topology, Prentice Hall (2000).
James R. Munkres, Topology, Pearson (2013).

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 a fundamental role in analysis.

In this course we introduce the general concept of a topological space (X,T) consisting of a set X together with a collection T of the so called open subsets of X satisfying certain natural axioms. 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.

Course Programme

FORMER Written exams: 29 May 2015, 18 August 2015.

History of Topology

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 (1934-2007).

The MacTutor History of Mathematics Archive