Loading presentation...

Present Remotely

Send the link below via email or IM


Present to your audience

Start remote presentation

  • Invited audience members will follow you as you navigate and present
  • People invited to a presentation do not need a Prezi account
  • This link expires 10 minutes after you close the presentation
  • A maximum of 30 users can follow your presentation
  • Learn more about this feature in our knowledge base article

Do you really want to delete this prezi?

Neither you, nor the coeditors you shared it with will be able to recover it again.


Make your likes visible on Facebook?

Connect your Facebook account to Prezi and let your likes appear on your timeline.
You can change this under Settings & Account at any time.

No, thanks

Kuratowski Closure Operator

Math Seminar 2012

Kimberly Nudelman

on 25 September 2012

Comments (0)

Please log in to add your comment.

Report abuse

Transcript of Kuratowski Closure Operator

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.
Full transcript