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Unformatted text preview: MGF3301 — practice questions
December 8, 2011
The ﬁnal exam will consist of two parts: the ﬁrst part will refer to material covered in the
two midterms; the second (longer) part will be based on the material we covered since the second
midterm: completeness of Dedekind cuts, cardinality, and topology. (The ﬁrst part will be worth
10 points; the second, 20 points.)
The following questions deal with the material covered in the second part. They are not meant
to look like exam questions, rather like study points. If you cannot conﬁdently answer all these
questions without looking up the class notes, then you are certainly not ready for the ﬁnal exam!
• What is the ‘least upper bound property’? Does Q satisfy it? Does D? Why?
• What is the ‘cardinality’ of a set? Can you give examples of sets with different cardinalities? • What does ‘uncountable’ mean?
• Why is there no ‘set of all sets’?
• What does the Cantor-Bernstein-Schr¨ der theorem say?
• What is a topology? What is a topological space? What is the ‘induced’ topology on a subset
of a topological space?
• How do you deﬁne the open sets in the standard topology on R? on Rn ?
• Can you give examples of sets that are closed, open, or neither in R with the standard topology? with the Zariski topology?
• What is the topological deﬁnition of ‘continuous function’?
• What is a ‘homeomorphism’? Can you give examples of spaces that are/are not homeomorphic? 1 ...
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This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.
- Fall '11