Exercises Exercises
2.
(a)
(b)
(c)
3.
Let and be metrically equivalent metric spaces. Show that and are topologically equivalent using the metric topologies.
4.
Prove Theorem 14.4 that if and are topological spaces, and is a function, then is continuous if and only if is a closed set in whenever is a closed set in
5.
(a)
Hint.
Exercise 9 could be helpful here.
(b)
(i)
(ii)
6.
(a)
Find a function that is continuous at exactly one point, or show that no such function exists.
(b)
Find a function that is continuous at exactly two points, or show that no such function exists.
(c)
Find a function that is continuous at exactly three points, or show that no such function exists.
7.
(a)
(b)
If we remove the condition that is continuous, must it still be the case that is homeomorphic to Prove your conjecture.
8.
Let be a nonempty set and let be a fixed element in Let be the particular point topology and the excluded point topology on That is
That the particular point and excluded point topologies are topologies is the subject of Exercise 9 and Exercise 10.
(a)
Let be a fixed point in Is the identity function defined by for all a homeomorphism from to Prove your answer.
(b)
9.
A topological space is embedded in a topological space if there is a homeomorphism from to some subspace of The homeomorphism is called an embedding.
(a)
Show that if is the open interval with the Euclidean metric topology, then can be embedded in the topological space with the Euclidean metric topology.
(b)
Show that there exist non-homeomorphic topological spaces and for which can be embedded in and can be embedded in
10.
Let be a two element set.
(a)
Find all of the distinct topologies on Be sure to explain how you know you have identified all of the topologies.
(b)
Determine the distinct homeomorphism classes of topological spaces on two elements. Justify your response.
11.
Let There are 29 distinct topologies on shown below. Determine the number of distinct homeomorphism classes for these 29 topologies and identify the elements of each homeomorphism class. Justify your answers.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
(q)
(r)
(s)
(t)
(u)
(v)
(w)
(x)
(y)
(z)
(aa)
(ab)
(ac)
the discrete topology
12.
Show that property is a topological property for each (See Chapter 13 for definitions of the separation axioms.)
13.
For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.
(a)
If is a continuous function between topological spaces and then for every open subset of is open in
(b)
(c)
If is a bijective function between topological spaces and and for every open subset of is open in then is a homeomorphism.
(d)
If and are topological space with the discrete topologies, and if is a bijection, then the spaces and are homeomorphic.