# e3key - M413 EXAM III Name There should be 5 distinct pages...

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M413 EXAM III Name There should be 5 distinct pages with a total of 9 problems. Unless otherwise noted, (M, d), (N, p) are arbitrary metric Ispaces, A C M, f : A -+ N. (3 pts. ea.) Define the following. 1 A <.4-~1 ~c. .. (b) A is compact c~~~ (c) A is sequentially compact ~ Cc-~~ A sc. .t6 S~1~L t~ .s~~ ~ ~ W I~ ~ ~; ~ A I If ..s~3~(;':tf": J M O,.oe.- ~ ~tA:i (d) A is connected ~~~r (e) <p : (a, b) -+ A is a contin~us path connecting x and y i~ A ;. ~ I.f "f (. ) -=- ".-' ~ lP(c. ..) -=- ~ ) 1

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2 (f) A is path connected I::tA ~ ,>( I~ G A -a Aa IB (i) f is continuous at Xo in A c~ '1<0 I ~ ~O"1 1 A ~.-t. .:tt f~ OM- OR. .pt~o ) A >' G , 2. (5 pts.) State the intermediate value theorem. ~ ,@ I d) h.L. tC. . ~ S' f t,I.,u. .,. ~ kCM 1C:IL.#. ~~t.c:t:-4 ~ ) a..J. ., ocA. k R ..ftI1. . ~A. 6 5:,+ c..(;-- ~ +(~) si-. ... """ c. -t (J<) ~ e,U. ~ n/1 t:s k ~f Q.'\A Coo +( z) =- I ~ X.l~ c A (g) open sets U and V separate A A c U \:IV AnunV:::' r;2f A(\v~rjJ 1r()U #1 (h) f is uniformly continuous on A 6.v~ {). ..c; ~ d '> ~ ~ I
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e3key - M413 EXAM III Name There should be 5 distinct pages...

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