Topology Prelim August 2010
1. Let Xand Ybe connected spaces. If Ais a proper subset of Xand Bis a proper subset of Ythen
X×Y−A×Bis connected. ( Ais a proper subset of Xif A⊂Xand A6=X.)
2. Let Xbe a first countable space.
(a) For any set A⊂Xand any point p∈X, show that p∈Aif and only if there is a sequence {pn}∞
n=1 in
Asuch that {pn}converges to p.
(b) Show that for any space Y, a map f:X−→ Yis continuous if and only if ftakes convergent sequences
in Xto convergent sequences in Y.
3. (a) If Xis a locally connected space then prove that the components of Xare open subsets of X.
(b) Let p:X−→ Ybe a quotient map. Show that if Xis locally connected, then Yis locally connected.
(Hint: If Cis a component of an open set Uof Y, show that p−1(C) is a union of components of p−1(U).)
4. Let Xbe a Hausdorff space. Suppose that {Aα|α∈ A} is a collection of compact, connected subsets of
Xsimply ordered by inclusion (that is, for each α, β ∈ A we have either Aα⊂Aβor Aβ⊂Aα). Prove
that ∩α∈AAαis nonempty and connected.
5. A continuous map f:X−→ Xis called a retraction of Xonto A=f(X) if f◦f=f. The image Aof f
is called a retract of X.
(a) Prove that any retract of a Hausdorff space is a closed set.
(b) Let a∈A. Show that f∗:π1(X, a)−→ π1(A, a) is surjective.
6. Let p:X−→ Ybe a covering map, where Xand Yare path connected and locally path connected, and
let x0∈p−1(y0). Prove the Unique Path Lifting Theorem: Suppose f: [0,1] −→ Yis any path with initial
point y0. Then there exists a unique lift ˜
f: [0,1] −→ Xof fsuch that ˜
f(0) = x0.
