Download 1. Basic Point Set Topology Consider Rn with its usual topology and

1. Basic Point Set Topology
Consider Rn with its usual topology and its usual vector space structure over R . Let
Lin(Rp , Rq ) stand for the set of linear maps from Rp to Rq and observe that it is isomorphic to
Rpq with respect to its natural structure as vector space over R .
PROBLEMS
I. Let X, Y be compact, connected, Hausdorff spaces.
(A) Prove that a local homeomorphism f : X −→ Y is a covering projection.
(B) Prove that an immersion h : Sn −→ Sn , where n ≥ 2 is a diffeomorphism. What happens if n = 1?
II. Let A, X, Y be topological spaces.
(A) If X is compact and Y is Hausdorff, prove that every continuous map f : X −→ Y is closed.
(B) If a surjective continuous map p : A −→ X is open or closed, prove that p is a quotient map. Give
counterexamples to demonstrate that there exist quotient maps that are neither open nor closed.
(C) Let q : A −→ X be a quotient map. Prove that g : X −→ Y is continuous iff g ◦ q : A −→ Y is continuous.
(D) Let r : A −→ X be an open, surjective map. Prove that h : X −→ Y is open iff h ◦ r : A −→ Y is open.
III. Let X be a topological space.
Given a group G, a (left) action of G on X (by homeomorphisms) is a continuous map (G assumed to have
the discrete topology) ρ : G × X −→ X such that (1) ρ(e, x) = x, (2) ρ(gh, x) = ρ(g, ρ(h, x)) for all g, h ∈ G and
x ∈ X. Clearly ρ(g, •) : X −→ X is a homeomorphism and ρ(g, •)−1 = ρ(g −1 , •) for each g ∈ G . In those cases
where there is no ambiguity as regards the choice of the action ρ, the notation ρ(g, x) is usually abbreviated into
gx. In the presence of a group action on X one introduces the equivalence relation ∼ with respect to which x ∼ y
if y = gx for some g ∈ G. By abuse of notation, the set of equivalence classes under ∼ is denoted by X/G . (A) If X/G has the quotient topology induced by the map q : X −→ X/G sending each x ∈ X into the
equivalence class containing x, prove that q is an open map.
(B) Write Tn , Sn , RP n respectively in the form Rn /G1 , (Rn+1 − {(0, 0, 0)})/G2 , (Rn+1 − {(0, 0, 0)})/G3 by
introducing suitable group actions.
IV. Let X be a topological space.
Let X be a topological space, R ⊆ X × X be an equivalence relation on X. Consider the quotient space X/R
and the quotient map q : X −→ X/R.
(A) If X/R is Hausdorff, prove that R is a closed subset of X × X.
(B) If q is an open map and R is a closed subset of X × X, prove that X/R is Hausdorff.
(C) Apply the above results to prove that Tn and RP n are Hausdorff spaces.
V. Remember that the (n + 1)-dimensional closed disc En+1 and the n-dimensional sphere Sn are defined by
En+1
=
{u ∈ Rn+1 | k u k≤ 1}
Sn
=
{u ∈ Rn+1 | k u k= 1} .
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(A) Let M, N be topological spaces and A, B ⊆ M be closed subsets with M = A ∪ B. Given continuous
functions f : A −→ N, g : B −→ N with f |A∩B = g|A∩B , prove that the function h : M −→ N defined by
f (m)
for
m∈A
h(m) =
g(m)
for
m∈B
is continuous.
(B) Let x : dom(x) ⊆op M −→ Rn be a chart. Given a closed set C ⊆ x(dom(x)), clearly x−1 (C) is closed
in dom(x). Is x−1 (C) is necessarily closed in M ? Suppose that K ⊆ x(dom(x)) is compact. If M is Hausdorff,
prove that x−1 (K) is closed in M.
(C) If M is an n-dimensional manifold, prove that there exists a surjective continuous map h : M −→ Sn .
(D) Remember that a topological space X is said to be homogeneous if for any a, b ∈ X there exists a
homeomorphism h : X −→ X such that h(a) = b. Prove that for any a, b ∈ Int En+1 , there exists a homeomorphism
h : En+1 −→ En+1 with h|Sn = IdSn , such that h(a) = b.
(E) Let M be a manifold. Prove that each m ∈ M has a neighbourhood U with the property that for each
a, b ∈ U there exists a homeomorphism h : M −→ M such that h(a) = b.
(F) Prove that a connected manifold is a homogeneous topological space.
(G) Give an example of a non-homogeneous manifold.
(H) Does there exist a connected, locally Euclidean, homogeneous topological space which is not Hausdorff ?
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