Mathematics, Faculty of Science

Sigmundur Gudmundsson


Spring 2017


Lecturer: Sigmundur Gudmundsson

Coordinates: Tuesdays and Fridays 13:15-15:00, lecture room 332B.

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 non-empty 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) generalising 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: 27 May 2016, 30 May 2017.

A History of Topology

A Few Heroes:

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