Download INTRODUCTION TO TOPOLOGY Homeworks 1) June 27: Define the

INTRODUCTION TO TOPOLOGY
Homeworks
1) June 27: Define the following topological space: a subset A of R is defined to be open if
A = ∅ or A = R or A = R − {p1 , . . . , pk }, where k ∈ N and p1 , . . . , pk ∈ R.
Show that this topological space is T0 and also T1 . Show that there is a sequence
in this topological space which has more than one limits. (So this space is not T2 .)
2) June 27: (a) Show that the projections πX : X × Y → X and πY : X × Y → Y are continuous. (The topological space X × Y is the direct product of the topological
spaces X and Y .)
(b) Show that a map f : Z → X × Y is continuous if and only if the compositions
πX ◦ f and πY ◦ f are continuous.
3) June 27: Show that a point p is in the closure of a set A if and only if every neighborhood
of p intersects A.
4) July 4: Let X be a topological space. Show that the two statements are equivalent:
(i) Only ∅ and X are the subsets of X which are clopen.
(ii) There are no disjoint, non-empty open subsets U and V of X such that
U ∪ V = X.
5) July 4: Show that in Q (its topology is the subspace topology in the Euclidean real line)
the only connected subsets are the singletons.
6) July 4: Show that the image of a path-connected space by a continuous map is pathconnected.
7) July 4: A collection of sets has the finite intersection property if the intersection of arbitrary
finitely many sets in the collection is non-empty. Show that if a topological space is
such that the intersection of any collection of closed sets with the finite intersection
property is non-empty, then the space is compact (i.e. any open cover has a finite
subcover).
8) July 11: Let (X, d) be a metric
S∞ space. For each i > 0, i ∈ N, let Ai be a finite 1/i-net.
Show that the set i=1 Ai is dense in X .
9) July 11: Show that in R2 with the usual metric d(x, y) = |x − y| a set A ⊂ R2 is bounded
(i.e. there is a constant K > 0 such that for all x, y ∈ A we have d(x, y) ≤ K ) if
and only if the set A is totally bounded (i.e. for every ε > 0 there is a finite ε-net).
10) July 11: Suppose X is a set and H is a collection of its subsets such that H has FIP and H
is maximal (i.e. if H ′ has FIP and H ⊆ H ′ , then H = H ′ ). Then it is easy to see
that if B ∈ H and B ⊂ C , then C ∈ H and if B1 , . . . , Bk ∈ H , then ∩ki=1 Bi ∈ H .
Show that if a set C intersects all members of H , then C ∈ H . (Hint: use the
previous two facts if necessary.)
11) July 18: Let (X, d) be a metric space. Define d′ : X × X → [0, ∞) to be d′ (x, y) =
min{1, d(x, y)}.
(i) Show that d′ is a metric on X .
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INTRODUCTION TO TOPOLOGY
(ii) Show that the identity map X → X is a homeomorphism, where the domain
has the metric d and the target has the metric d′ . Hint: use sequential
continuity.
Let f1 , f2 : [0, 1] → X and g1 , g2 : [0, 1] → X be loops in X with basepoint x0 .
Suppose f1 is homotopic to f2 and g1 is homotopic to g2 . Show that the concatenation f1 g1 is homotopic to the concatenation f2 g2 .
Show that an f : X → Y continuous map, where f (x0 ) = y0 for the basepoints
x0 ∈ X and y0 ∈ Y , induces a group homomorphism f∗ : π1 (X, x0 ) → π1 (Y, y0).
(Hint: if you have a loop l : [0, 1] → X or a homotopy F : [0, 1] × [0, 1] → X , then
f ◦ l is a loop in Y and f ◦ F is a homotopy in Y .)
Take the rectangle [−1, 1]×[−1, 1]. Glue together its boundary by the rule (−1, t) =
(1, −t) and (s, −1) = (−s, 1). Denote the resulting topological space by X . Take
the Möbius band and a disk. Glue the disk along its boundary (which is a circle)
to the boundary of the Möbius band (which is also a circle) by identifying the two
circles. Denote the resulting topological space by Y . Show that X and Y are
homeomorphic.
Show that if we glue together two Möbius bands along their boundaries, then we
get the Klein bottle.
Show a covering map from a simply connected space onto the “figure eight”.