Kimberly Nudelman

September 25, 2012 The Kuratowski Closure Operator What is Topology? The Kuratowski

Closure Operator Examples! Objectives: What is topology? - definition

- topological space & open sets

- closed sets

- the closure of a set

- theorem 1

- last but definitely not least...why is it important? Brief history Kuratowski's closure operator - proof

- examples "... the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects..." Definitions: --mathworld.wolfram.com/topology.html "The study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures" --google.com Topological Space Closed sets: The closure of a set: We have: Theorem 1: Theorem 2: The proof: By our proof: Kuratowski's View: Define a collection: given: need to show: NTS: Note Lemma 1: Lemma 1: NTS: NTS: Traditional View: Perspectives... NTS: We have: By Theorem 1: In Perspective... History: NTS: NTS: NTS: (T.T.T.T.B.A.S.I.) Kazimierz Kuratowski was born in Warsaw on February 2, 1896 During WWII, he gave lectures at the underground university in Warsaw, since higher education for Poles was forbidden under German occupation In 1921, he received his Ph.D. in mathematics His thesis consisted of two parts, one of which was devoted to an axiomatic depiction of topology due to an introduction of the closure axioms --

Sur la notion de l'ensemble fini, "Fundamenta Mathematicae," 1920 Played a large role in the re-development of the scientific community in Poland after WWII closed open We have: By Theorem 1: Trivial: Discrete: Co-finite: Co-countable: References: O'Connor, J. J., and E. F. Robertson. "Kazimierz Kuratowski." Kuratowski Biography.

JOC/EFR, Feb. 2000. Web. 16 Sept. 2012. <http://www-history.mcs.st-

and.ac.uk/Biographies/Kuratowski.html>. "Kazimierz Kuratowski." Wikipedia. Wikimedia Foundation, 09 Nov. 2012. Web. 16

Sept. 2012. <http://en.wikipedia.org/wiki/Kazimierz_Kuratowski>. Weisstein, Eric W. "Topology." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/Topology.html Munkres, J. R. Topology, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000. Poincaré, Henri. "Chapter II: Mathematical Magnitude and Experiment". Science and

Hypothesis. "Mathematicians do not study objects, but the relations between objects..." --Henri Poincaré Q.E.D.

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# Kuratowski Closure Operator

Math Seminar 2012

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