Math 500: Topology

Homework 7
Lawrence Tyler Rush
<me@tylerlogic.com>
January 12, 2013
http://coursework.tylerlogic.com/courses/math500/homework07
(1)
ℝn
(a)
connected: Yes, it is the finite product of a connected set, ℝ.
(b)
path connected: Yes, the straight line between any two points will be continuous.
(c)
metrizable: Yes, using the euclidean or square metric. Munkres Theorem 20.3
(d)
compact: No, ℝ is not even compact, so ℝn is not.
(e)
T1: Yes, it’s metrizable.
(f)
T2: Yes, it’s metrizable.
(g)
T3: Yes, it’s metrizable.
(h)
T4: Yes, it’s metrizable.
(i)
first-countable: Yes, it’s metrizable.
(j)
second-countable: Yes, it is the finite product of the second countable set ℝ.
(k)
locally Euclidean: Yes, it is ℝn.
(2)
ℝd
(a)
connected: No, every set is clopen.
(b)
path connected: No, because its not connected.
(c)
metrizable: Yes, given by the discrete metric.
(d)
compact: No, this is finer than ℝ and ℝ is not compact.
(e)
T1: Yes, it’s metrizable.
(f)
T2: Yes, it’s metrizable.
(g)
T3: Yes, it’s metrizable.
(h)
T4: Yes, it’s metrizable.
(i)
first-countable: Yes, it’s metrizable.
(j)
second-countable: No, it is finer than ℝℓ and that is not second-countable.
(k)
locally Euclidean: Yes, any open set of ℝ will be an open set here. So every open neighborhood of any element of ℝd has a neighborhood which is open in ℝ.
(3)
ℝℓ
(a)
connected: No, [-∞,x) and [x,+∞) are open sets that separate the space for any x ∈ ℝℓ.
(b)
path connected: No, because it is not connected.
(c)
metrizable: Yes, in each lower limit basis element we can fit an open ball of the euclidean metric, and vice versa.
(d)
compact: No, {[-n,n) | n ∈ ℤ+} has no finite subcover.
(e)
T1: Yes, it’s metrizable.
(f)
T2: Yes, it’s metrizable.
(g)
T3: Yes, it’s metrizable.
(h)
T4: Yes, it’s metrizable.
(i)
first-countable: Yes, it’s metrizable.
(j)
second-countable: No, section 30 example 3.
(k)
locally Euclidean: Yes, locally 1-euclidean. For any x ∈ ℝℓ the set ∪n[x - 1 + 1n,x + 1) contains x, is open in ℝℓ, and equals the open set of ℝ, (x - 1,x + 1).
(4)
ℝfc
(a)
connected: Yes, a non-trivial clopen set would need to be finite and infinite, which isn’t possible. Thus there are no non-trivial clopen sets.
(b)
path connected: Yes, the constant function on ℝ will satisfy the requirements.
(c)
metrizable: No, not Hausdorff.
(d)
compact: Yes, any element of an open cover, A, will have a finite number of elements not contained in it, so we only need to select a finite number of other elements from A to cover the entire space.
(e)
T1: Yes, one point sets are finite, so their complements are open, so they are closed.
(f)
T2: No, since no open sets in ℝfc are disjoint.
(g)
T3: No, not Hausdorff.
(h)
T4: No, not Hausdorff.
(i)
first-countable: No. Assume for contradiction that {Ai} is a countable basis at x. Then U = ∪i{ℝ\Ai} is a countable collection of points since it is a countable union of finite sets. Then for any y ∈ ℝ \ U (ℝ \ U is nonempty since ℝ is uncountable), ℝ \{y} is open in ℝfc but contains no Ai.
(j)
second-countable: No, since it is not first countable.
(k)
locally Euclidean: Yes, every neighborhood of a point x is an open set in ℝ since finite sets are closed in ℝ. Thus each x has a neighborhood homeomorphic to an open set of ℝ.
(5)
S1
(a)
connected: Yes, it is path connected.
(b)
path connected: Yes, Section 24 example 5.
(c)
metrizable: Yes, the magnitude of the acute angle between two points in the set.
(d)
compact: Yes, closed and bounded subset of ℝ2, so compact by Heine-Borel.
(e)
T1: Yes, it’s metrizable.
(f)
T2: Yes, it’s a subspace of the Hausdorff space ℝ2.
(g)
T3: Yes, it’s normal.
(h)
T4: Yes, it’s Hausdorff and compact.
(i)
first-countable: Yes, it’s metrizable.
(j)
second-countable: Yes, it’s a subspace of a second countable space, ℝ2.
(k)
locally Euclidean: Yes, it’s a subspace of a ℝ2.
(6)
ℝ × ℝ, dictionary
(a)
connected: Yes, it is path connected.
(b)
path connected: Yes, the straight line between any two points will be continuous.
(c)
metrizable: Yes, Section 50 supplementary exercise 4.
(d)
compact: No, since it is Hausdorff, being compact would imply that it is an m-manifold, but it is not due to Section 50 exercise 2.
(e)
T1: Yes, this is locally euclidean.
(f)
T2: Yes, this is normal.
(g)
T3: Yes, this is normal.
(h)
T4: Yes, this is an order topology.
(i)
first-countable: Yes, it’s metrizable.
(j)
second-countable: No, Section 50 exercise 4.
(k)
locally Euclidean: Yes, it is 1-euclidean.
(7)
ℝω, product topology
(a)
connected: Yes, Section 23 example 7.
(b)
path connected: ???
(c)
metrizable: Yes, Munkres Theorem 20.5.
(d)
compact: No, ℝ is not compact.
(e)
T1: Yes, it’s metrizable.
(f)
T2: Yes, it’s metrizable.
(g)
T3: Yes, it’s metrizable.
(h)
T4: Yes, it’s metrizable.
(i)
first-countable: Yes, it’s metrizable.
(j)
second-countable: Yes, since ℝ is second countable, by Munkres Theorem 30.2.
(k)
locally Euclidean: Yes, since it’s metrizable.
(8)
TSC
(a)
connected: Yes, Section 24 example 7.
(b)
path connected: No, Section 24 example 7.
(c)
metrizable: Yes, it’s regular and second countable. (Urysohn’s metrization theorem).
(d)
compact: Yes, it is a closed and bounded subset of ℝ2.
(e)
T1: Yes, it’s metrizable.
(f)
T2: Yes, it’s a subspace of the Hausdorff space ℝ2.
(g)
T3: Yes, it’s normal.
(h)
T4: Yes, it’s compact and Hausdorff.
(i)
first-countable: Yes, it’s metrizable.
(j)
second-countable: Yes, it’s a subspace of the second countable space ℝ2.
(k)
locally Euclidean: Yes, it’s a subspace of ℝ2.