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UID:www.tcs.tifr.res.in/event/754
DTSTAMP:20230914T125937Z
SUMMARY:Alternate Characterisations of Compact Spaces
DESCRIPTION:Speaker: Sayantan  Chakraborty\n\nAbstract: \nThe usual idea of
  compactness of a space is that if the space has an open cover then it has
  a finite subcover which is not very intuitive. In this talk we attempt to
  look at some equivalent characterisations of compact spaces. At first we 
 prove some important connections between compactness \, totally boundednes
 s and sequential compactness using a series of theorems that begin by prov
 ing  that every compact metric space has the Bolzano Weierstrass property
 \, move on to the Lebesgue covering lemma and then end by proving that eve
 ry sequentially compact metric space is totally bounded and is compact.\nT
 he second part of the talk attempts to apply these ideas to a proof of Asc
 olis theorem the statement of which is given : If $X$ is a a compact metri
 c space then a closed subspace of $C(X\,R) or C(X\,C)$ is compact iff  it
  is bounded and equicontinuous. Along the way. We develop certain importan
 t ideas such as a metric space is compact iff it is complete and totally b
 ounded. We conclude by mentioning some important applications of Ascolis t
 heorem\, for example Prokhorov's theorem in probability.\n
URL:https://www.tcs.tifr.res.in/web/events/754
DTSTART;TZID=Asia/Kolkata:20170217T160000
DTEND;TZID=Asia/Kolkata:20170217T173000
LOCATION:A-201 (STCS Seminar Room)
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