## 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 $$**R**,
in the plan $$**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:** 29 May 2015, 18 August 2015.

**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).

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