Connectedness Continued
Adrian Down 16779577
August 9, 2005
1 Last Time
Wanted to generalize theorems about continuous functions to general metric
spaces. Had to generalize closed intervals, which is compactness. From last
time, Ecompact and fcontinuous ⇒f(E) is compact. Also, Econnected
and fcontinuous ⇒f(E) is connected. Continuous functions preserve con-
nectedness and compactness.
Recall: 1) Eis connected if Eis not disconnected. Eis disconnected if ∃
open sets U1, U2such that
U1∩U2=∅
E⊆U1∪U2
E∩U16=∅
E∩U26=∅
2) Uopen in S2and f:S1→S2is continuous, then f−1(U) is open in S1.
Also, Eclosed in S2, f −1(E) is closed in S1,f−1(E) = S1r(f−1(S2rE)).
3) E⊆Sis path-connected if ∀x, y ∈E, ∃a path from xto yin E, i.e.
γ: [a, b]→S,γcontinuous, γ(a) = x, γ(b) = y, γ (t)∈E, a ≤t≤b.
Notes: i) γ[a, b] is compact in Sand connected.
ii) If ∃a path from xto yand a path from yto z, then ∃a path from x
to z.
Let γ1[a, b]→S,γ1(a) = x, γ1(b) = y,γ2[b, d]→S, γ2(c) = y, γ2(d) = z.
γ2can always be arranged to go from bby change of variables. Now let
γ: [a, b]→S, γt=(γ1(t)a≤t≤b
γ2(t)b≤t≤d
Theorem: Eis path connected ⇒Eis connected.
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