Exercises

Exercises Exercises

1.

Let \((X,d_X)\) and \((Y,d_Y)\) be metrically equivalent metric spaces, and let \(f:X \to Y\) be a bijection such that

\begin{equation*} d_X(x,y) = d_Y(f(x),f(y)) \end{equation*}

for all \(x,y \in X\text{.}\) Prove that

\begin{equation*} d_Y(u,v) = d_X(f^{-1}(u), f^{-1}(v)) \end{equation*}

for all \(u\) and \(v\) in \(Y\text{.}\)

2.

Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces, and let \(f : X \to Y\) be a homeomorphism. Let \(A\) be a subset of \(X\text{.}\)

(a)

If \(x\) is a limit point of \(A\text{,}\) must \(f(x)\) be a limit point of \(f(A)\text{?}\) Prove your answer.

(b)

If \(x\) is an interior point of \(A\text{,}\) must \(f(x)\) be an interior point of \(f(A)\text{?}\) Prove your answer.

(c)

If \(x\) is a boundary point of \(A\text{,}\) must \(f(x)\) be a boundary point of \(f(A)\text{?}\) Prove your answer.

3.

Let \((X, d_X)\) and \((Y, d_Y)\) be metrically equivalent metric spaces. Show that \(X\) and \(Y\) are topologically equivalent using the metric topologies.

4.

Prove Theorem 14.4 that if \((X,\tau_X)\) and \((Y, \tau_Y)\) are topological spaces, and \(f: X \to Y\) is a function, then \(f\) is continuous if and only if \(f^{-1}(C)\) is a closed set in \(X\) whenever \(C\) is a closed set in \(Y\text{.}\)

5.

Let \((X, \tau_X)\text{,}\) \((Y, \tau_Y)\text{,}\) and \((Z, \tau_Z)\) be topological spaces.

(a)

Let \(f: X \to Y\) and \(g : Y \to Z\) be continuous functions. Prove that \(g \circ f : X \to Z\) is a continuous function.

Hint.

Exercise 9 could be helpful here.

(b)

Let \(f: (X, \tau_X) \to (Y, \tau_Y)\) be a homeomorphism. Let \(h\) be a function from \((Y, \tau_Y)\) to \((Z, \tau_Z)\) and let \(k\) be a function from \((Z, \tau_Z)\) to \((X, \tau_X)\text{.}\)

(i)

Prove that \(h\) is continuous if and only if \(h \circ f\) is continuous.

(ii)

Prove that \(k\) is continuous if and only if \(f \circ k\) is continuous.

6.

Let \(X = \{a,b,c,d\}\) with topology \(\tau = \{\emptyset, \{a\}, \{b\}, \{a,b\}, \{b,d\}, \{a,b,d\}, X\}\text{.}\)

(a)

Find a function \(f: X \to X\) that is continuous at exactly one point, or show that no such function exists.

(b)

Find a function \(f: X \to X\) that is continuous at exactly two points, or show that no such function exists.

(c)

Find a function \(f: X \to X\) that is continuous at exactly three points, or show that no such function exists.

7.

Consider \(\R\) and \(\R^2\) equipped with the Euclidean topology. Let \(f : \R \to \R\) be a function and let

\begin{equation*} \Gamma_f = \{(x,f(x)) \mid x \in \R\} \end{equation*}

be the graph of \(f\text{.}\) Note that \(\Gamma_f\) is a subspace of \(\R^2\) and is a topological space using the subspace topology.

(a)

Show that if \(f\) is a continuous function, then \(\Gamma_f\) is homeomorphic to \(\R\text{.}\)

(b)

If we remove the condition that \(f\) is continuous, must it still be the case that \(\Gamma_f\) is homeomorphic to \(\R\text{?}\) Prove your conjecture.

8.

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.

(a)

Let \(p\) be a fixed point in \(\R\text{.}\) Is the identity function \(i: \R \to \R\) defined by \(i(x) = x\) for all \(x \in \R\) a homeomorphism from \((\R, \tau_p)\) to \((\R, \tau_{\overline{p}})\text{?}\) Prove your answer.

(b)

Is \((\R, \tau_{p})\) homeomorphic to \((\R, \tau_{\overline{p}})\) with the specific point \(p=0\text{?}\) Prove your answer.

9.

A topological space \(X\) is embedded in a topological space \(Y\) if there is a homeomorphism from \(X\) to some subspace of \(Y\text{.}\) The homeomorphism is called an embedding.

(a)

Show that if \(X\) is the open interval \((0,1)\) with the Euclidean metric topology, then \(X\) can be embedded in the topological space \(\R\) with the Euclidean metric topology.

(b)

Show that there exist non-homeomorphic topological spaces \(A\) and \(B\) for which \(A\) can be embedded in \(B\) and \(B\) can be embedded in \(A\text{.}\)

10.

Let \(X = \{a,b\}\) be a two element set.

(a)

Find all of the distinct topologies on \(X\text{.}\) 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 \(X = \{a,b,c\}\text{.}\) There are 29 distinct topologies on \(X\text{,}\) 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)

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

(b)

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

(c)

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

(d)

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

(e)

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

(f)

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

(g)

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

(h)

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

(i)

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

(j)

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

(k)

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

(l)

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

(m)

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

(n)

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

(o)

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

(p)

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

(q)

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

(r)

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

(s)

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

(t)

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

(u)

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

(v)

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

(w)

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

(x)

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

(y)

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

(z)

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

(aa)

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

(ab)

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

(ac)

the discrete topology

12.

Show that property \(T_i\) is a topological property for each \(i\text{.}\) (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 \(f : X \to Y\) is a continuous function between topological spaces \(X\) and \(Y\text{,}\) then for every open subset \(U\) of \(X\text{,}\) \(f(U)\) is open in \(Y\text{.}\)

(b)

If \(\tau_{FC}\) is the finite complement topology, then \(f(x) = x^2\) mapping \((\R, \tau_{FC})\) to \((\R, \tau_{FC})\) is continuous.

(c)

If \(f : X \to Y\) is a bijective function between topological spaces \(X\) and \(Y\text{,}\) and for every open subset \(U\) of \(X\text{,}\) \(f(U)\) is open in \(Y\text{,}\) then \(f\) is a homeomorphism.

(d)

If \(X\) and \(Y\) are topological space with the discrete topologies, and if \(f: X \to Y\) is a bijection, then the spaces \(X\) and \(Y\) are homeomorphic.

(e)

Let \(S\) be a set and let \(R_1\) and \(R_2\) be equivalence relations on \(S\text{.}\) Then \(R = R_1 \cap R_2\) is also an equivalence relation on \(S\text{.}\)

(f)

Let \(S\) be a set and let \(R_1\) and \(R_2\) be equivalence relations on \(S\text{.}\) Then \(R = R_1 \cup R_2\) is also an equivalence relation on \(S\text{.}\)