## Mathematics, Faculty of Science## Sigmundur Gudmundsson## Topology## Spring 2016 |

**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 $$**R**,
in the plan $$**R**^{2} or in the general n-dimensional space
$$**R**^{n} 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.

**FORMER Written exams:** 27 May 2016, 18 August 2016.

**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