Exercises

Exercises Exercises

1.

Recall from Definition 12.13 that if \(\tau_1\) and \(\tau_2\) are two topologies on a set \(X\) such that \(\tau_1 \subseteq \tau_2\text{,}\) then \(\tau_1\) is said to be a coarser (or weaker) topology than \(\tau_2\text{,}\) or \(\tau_2\) is a finer (or stronger) topology than \(\tau_1\text{.}\) In this exercise we explore the question of whether compactness is a property that is passed from weaker to stronger topologies or from stronger to weaker. Let \(\tau_1\) and \(\tau_2\) be two topologies on a set \(X\text{.}\) If \(\tau_1 \subseteq \tau_2\text{,}\) what does connectedness under \(\tau_1\) or \(\tau_2\) imply, if anything, about compactness under the other topology? Justify your answers.

2.

Let \(f: [a,b] \to \R\) be a continuous function from a closed interval into the reals. Let \(U = f(u)\) and \(V = f(v)\) be such that \(U \leq f(x) \leq V\) for all \(x \in [a,b]\text{.}\) Prove that there is a \(w\) between \(u\) and \(v\) such that \(f(w)(b-a) = \int_a^b f(t) \, dt\text{.}\)

3.

Let \(A\) be a connected subset of a topological space \(X\text{.}\) Prove or disprove:

(a)

\(\Int(A)\) is connected

(b)

\(\overline{A}\) is connected

(c)

\(\Bdry(A)\) is connected

4.

Let \(X = \R\) with the finite complement topology. We have shown that every subset of any topological space with the finite complement topology is compact. Now find all of the connected subsets of \(X\text{.}\) Prove your result.

5.

Give examples, with justification, of subsets \(A\) and \(B\) of a topological space to illustrate each of the following, or explain why no such example exists:

(a)

\(A\) and \(B\) are connected, but \(A \cap B\) is disconnected

(b)

\(A\) and \(B\) are connected, but \(A \setminus B\) is disconnected

(c)

\(A\) and \(B\) are disconnected, but \(A \cup B\) is connected

(d)

\(A\) and \(B\) are connected and \(A \cap B \neq \emptyset\text{,}\) but \(A \cup B\) is disconnected.

(e)

\(A\) and \(B\) are connected and \(\overline{A} \cap \overline{B} \neq \emptyset\text{,}\) but \(A \cup B\) is disconnected.

6.

Let \(f: S^1 \to \R\) be a continuous function. Show that there is a point \(x \in S^1\) with \(f(x) = f(-x)\text{.}\)

7.

Let \(a, b \in \R\) with \(a \lt b\text{.}\) Explain why no two of the sets \((a,b)\text{,}\) \((a,b]\text{,}\) and \([a,b]\) homeomorphic.

8.

Let \(K = \left\{\frac{1}{k} \mid k \text{ is a positive integer} \right\}\text{.}\) Let \(\B\) be the collection of all open intervals of the form \((a,b)\) and all sets of the form \((a,b) \setminus K\text{,}\) where \(a \lt b\) are real numbers as in Example 13.13. Let \(\tau_K\) be the topology generated by \(\B\text{.}\) Show that \((\R, \tau_K)\) is a connected space.

9.

Even though \(X = (0,1) \cup (1,2)\) is not a connected space, if \(x\) is any element in \(X\) then we can surround \(x\) with a connected subset of \(X\text{.}\) This is the idea of local connectedness.

Definition 18.16.

A topological space \(X\) is locally connected at a point \(x \in X\) if every neighborhood \(U\) of \(x\) contains an open connected neighborhood of \(x\text{.}\) A topological space \(X\) is locally connected if \(X\) is locally connected at each point in \(X\text{.}\)

(a)

Give an example of a locally connected space that is not connected.

(b)

It would be reasonable to believe that a connected space is locally connected. However, that is not the case. Consider the space \(X = A \cup B\) as a subspace of \(\R^2\) with the standard Euclidean metric topology, where \(A = \{(x,y) \mid x \text{ is irrational and } 0 \leq y \leq 1\}\) and \(B = \{(x,y) \mid x \text{ is rational and } -1 \leq y \leq 0\}\text{.}\)

(i)

Explain why \(X\) is connected.

(ii)

Show that \(X\) is not locally connected.

Hint.

Let \(x\) be a point not on the \(x\)-axis and find an open ball around \(x\) that doesn’t intersect the \(x\)-axis.

(c)

Prove that a topological space \(X\) is locally connected if and only if for every open set \(O\) in \(X\text{,}\) the connected components of \(O\) are open in \(X\text{.}\)

10.

Let \(A\) and \(B\) be nonempty subsets of a topological space \(X\text{.}\)

(a)

Prove that \(A \cup B\) is disconnected if \((\overline{A} \cap B ) \cup (A \cap \overline{B}) = \emptyset\text{.}\)

(b)

Prove that \(X\) is connected if and only if for every pair of nonempty subsets \(A\) and \(B\) of \(X\) such that \(X = A \cup B\) we have \((\overline{A} \cap B ) \cup (A \cap \overline{B}) \neq \emptyset\text{.}\)

11.

Give examples of the following.

(a)

A topological space with exactly one cut point.

(b)

A topological space with exactly two cut points.

(c)

A topological space with infinitely many cut points.

(d)

A topological space with no cut points.

12.

Let \(a, b \in \R\) with \(a \lt b\text{.}\) Prove that a homeomorphism \(f: [a,b] \to [a,b]\) carries end points into end points.

13.

Let \(X\) and \(Y\) be a topological spaces.

(a)

Assume that \(X\) and \(Y\) homeomorphic spaces. Prove that there is a one-to-one correspondence between the connected components of \(X\) and the connected components of \(Y\text{.}\)

(b)

Assume that \(X\) and \(Y\) homeomorphic spaces. Prove that there is a one-to-one correspondence between the set of cut points of \(X\) and the set of cut points of \(Y\text{.}\)

(c)

Consider each letter in the statement as a topological space with the standard Euclidean metric topology.

TOPOLOGY IS NEAT

Group the letters in the statement into disjoint homeomorphism classes. Explain in detail the reasons for your groupings.

14.

Let \((X, \tau)\) be a topological space.

(a)

Prove that \(X\) is disconnected if and only if \(X\) has a proper subset that is both open and closed.

(b)

Prove that \(X\) is disconnected if and only if there is a continuous function from \(X\) onto a discrete two-point topological space.

15.

Let \(X\) be the set of real numbers.

(a)

Consider \(X\) with the topology \(\tau_1 = \{\emptyset, [0,1], X\}\text{.}\) Prove or disprove: \(X\) is connected.

(b)

Consider \(X\) with the topology \(\tau_2 = \{U \subseteq X \mid 0 \in U\} \cup \{\emptyset\}\text{.}\)

(i)

Prove or disprove: \(X\) is connected.

(ii)

Prove or disprove: \(X \setminus \{0\}\) is connected.

16.

Let \(X\) be a nonempty set and let \(p\) be a fixed element in \(X\text{.}\) Let \(\tau_p\) be the particular point topology and \(\tau_{\overline{p}}\) the excluded point topology on \(X\text{.}\) That is

\(\tau_{p}\) is the collection of subsets of \(X\) consisting of \(\emptyset\text{,}\) \(X\text{,}\) and all of the subsets of \(X\) that contain \(p\text{.}\)
\(\tau_{\overline{p}}\) is the collection of subsets of \(X\) consisting of \(\emptyset\text{,}\) \(X\text{,}\) and all of the subsets of \(X\) that do not contain \(p\text{.}\)

That the particular point and excluded point topologies are topologies is the subject of Exercise 9 and Exercise 10. Determine, with proof, the connected subsets of \(X\) when

(a)

\(X\) has the particular point topology \(\tau_p\)

(b)

\(X\) has the excluded point topology \(\tau_{\overline{p}}\text{.}\)

17.

Let \((X, \tau)\) be a topological space and \(A\) a connected subset of \(X\text{.}\)

(a)

Show that if \(X\) is Hausdorff, then \(A'\) is connected.

(b)

Let \((X, \tau) = (\R, \tau_0)\text{,}\) where \(\tau_0\) is the particular point topology on \(X\text{.}\) Explain why \(A = \Z\) is a connected subset of \(X\text{.}\) Find \(\Z'\) in \((\R, \tau_0)\text{.}\) Is it true that in any topological space, if \(A\) is connected, then so is \(A'\text{?}\) Explain. (See Exercise 16.)

18.

Let \(X\) be a topological space. Prove each of the following.

(a)

Each \(a \in X\) is an element of exactly one connected component \(C_a\) of \(X\text{.}\)

(b)

A component \(C_a\) contains all connected subsets of \(X\) that contain \(a\text{.}\) Thus, \(C_a\) is the union of all connected subsets of \(X\) that contain \(a\text{.}\)

(c)

If \(a\) and \(b\) are in \(X\text{,}\) then either \(C_a = C_b\) or \(C_a \cap C_b = \emptyset\text{.}\)

(d)

Every connected subset of \(X\) is a subset of a connected component.

(e)

Every connected component of \(X\) is a closed subset of \(X\text{.}\)

(f)

The space \(X\) is connected if and only if \(X\) has exactly one connected component.

19.

Let \(X\) be a topological space with only a finite number of connected components. Show that each component of \(X\) is open.

20.

Let \(X\) and \(Y\) be connected spaces with \(f : X \to Y\) a continuous function. Is it the case that if \(S\) is a cut set of \(X\text{,}\) then \(f(S)\) is a cut set of \(Y\text{?}\) Prove your answer.

21.

Let \(X = \{a,b,c,d\}\text{.}\) There are 355 distinct topologies on \(X\text{,}\) but they fit into the 33 distinct homeomorphism classes listed below. The list is ordered by decreasing number of singleton sets in the topology, and, when that is fixed, by increasing number of two-point subsets and then by increasing number of three-point subsets. Under which topologies is \(X\) connected? Prove your answer.

(a)

the discrete topology

(b)

\(\{\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}, X\}\)

(c)

\(\{\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}, \{a,b,d\}, X\}\)

(d)

\(\{\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,d\}, \{a,b,c\}, \{a,b,d\}, \{a,c,d\}, X\}\)

(e)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, X\}\)

(f)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,b,c\}, X\}\)

(g)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,c,d\}, X\}\)

(h)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,b,c\}, \{a,b,d\}, X\}\)

(i)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,c\}, \{a,b,c\}, X\}\)

(j)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,c\}, \{a,b,c\}, \{a,c,d\}, X\}\)

(k)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,c\}, \{a,b,c\}, \{a,b,d\}, X\}\)

(l)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{c,d\}, \{a,c,d\}, \{b,c,d\}, X\}\)

(m)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,c\}, \{a,d\}, \{a,b,c\}, \{a,b,d\}, X\}\)

(n)

\(\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,c\}, \{a,d\}, \{a,b,c\}, \{a,b,d\}, \{a,c,d\}, X\}\)

(o)

\(\{\emptyset, \{a\}, X\}\)

(p)

\(\{\emptyset, \{a\}, \{a,b\}, X\}\)

(q)

\(\{\emptyset, \{a\}, \{a,b\}, \{a,b,c\}, X\}\)

(r)

\(\{\emptyset, \{a\}, \{b,c\}, \{a,b,c\}, X\}\)

(s)

\(\{\emptyset, \{a\}, \{a,b\}, \{a,c,d\}, X\}\)

(t)

\(\{\emptyset, \{a\}, \{a,b\}, \{a,b,c\}, \{a,b,d\}, X\}\)

(u)

\(\{\emptyset, \{a\}, \{b,c\}, \{a,b,c\}, \{b,c,d\}, X\}\)

(v)

\(\{\emptyset, \{a\}, \{a,b\}, \{a,c\}, \{a,b,c\}, X\}\)

(w)

\(\{\emptyset, \{a\}, \{a,b\}, \{a,c\}, \{a,b,c\}, \{a,b,d\}, X\}\)

(x)

\(\{\emptyset, \{a\}, \{c,d\}, \{a,b\}, \{a,c,d\}, X\}\)

(y)

\(\{\emptyset, \{a\}, \{a,b\}, \{a,c\}, \{a,d\}, \{a,b,c\}, \{a,b,d\}, \{a,c,d\}, X\}\)

(z)

\(\{\emptyset, \{a\}, \{a,b,c\}, X\}\)

(aa)

\(\{\emptyset, \{a\}, \{b,c,d\}, X\}\)

(ab)

\(\{\emptyset, \{a,b\}, X\}\)

(ac)

\(\{\emptyset, \{a,b\}, \{c,d\}, X\}\)

(ad)

\(\{\emptyset, \{a,b\}, \{a,b,c\}, X\}\)

(ae)

\(\{\emptyset, \{a,b\}, \{a,b,c\}, \{a,b,d\}, X\}\)

(af)

\(\{\emptyset, \{a,b,c\}, X\}\)

(ag)

\(\{\emptyset, X\}\)

22.

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 \(A\) is a connected subset of a topological space \(X\) with \(|A| \geq 2\text{,}\) then every point of \(A\) is a limit point of \(A\text{.}\)

(b)

If \(A\) is a compact subspace of a Hausdorff space, then \(A\) is connected.

(c)

If \(A\) is a connected subspace of a Hausdorff space, then \(A\) is compact.

(d)

Every subset of a topological space with the discrete topology is disconnected.

(e)

The set \(\{a,b\}\) is a connected component of the topological space \(X = \{a,b,c,d\}\) with topology

\begin{equation*} \tau = \{\emptyset, \{a\}, \{a,b\}, \{c\}, \{d\}, \{c,d\}, \{a,c,d\}, \{a,c\}, \{a,d\}, \{a,b,c\}, \{a,b,d\}, X\}\text{.} \end{equation*}

(f)

The sets \(U = \{a,c,d\}\) and \(V = \{a,b,c\}\) form a separation of the set \(A = \{c,d\}\) in the topological space \(X = \{a,b,c,d\}\) with topology

\begin{equation*} \tau = \{\emptyset, \{a\}, \{a,b\}, \{c\}, \{d\}, \{c,d\}, \{a,c,d\}, \{a,c\}, \{a,d\}, \{a,b,c,\}, \{a,b,d\}, X\}\text{.} \end{equation*}

(g)

The connected topological space \(X = \{a,b,c,d\}\) with topology

\begin{equation*} \tau = \{\emptyset, \{a\}, \{b,c\}, \{a,b,c\}, X\} \end{equation*}

has no cut points.