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Topology: An Inquiry-Based Approach

Steven Schlicker

Exercises Exercises

1.

(a)

Let \((Y_1, \tau_1)\) and \((Y_2, \tau_2)\) be topological spaces, where \(Y_1 = \{a,b,c\}\) with \(\tau_1 = \{\emptyset, \{a\}, \{b,c\}, Y_1\}\) and \(Y_2 = \{1,2\}\) with \(\tau_2 = \{\emptyset, \{1\}, Y_2\}\text{.}\) Find all of the sets of the form
\begin{equation*} \CS = \{\pi_1^{-1}(O_1) \mid O_1 \text{ is open in } Y_1\} \cup \{\pi_2^{-1}(O_2) \mid O_2 \text{ is open in } Y_2\} \end{equation*}
and verify that these sets generate the product topology on \(Y_1 \times Y_2\text{.}\)

(b)

Let \(X_{i}\) for \(i\) from \(1\) to \(n\) be topological spaces and let \(X = \Pi_{i=1}^n X_i\text{.}\) Show that the collection
\begin{equation*} \CS = \bigcup_{i=1}^n \{\pi_i^{-1}(O_i) \mid O_i \text{ is open in } X_i\} \end{equation*}
is a subbasis for the box topology on \(X\text{.}\)

2.

Let \(X\) be the set of real numbers with the standard Euclidean metric topology and let \(Y\) be the real numbers with the discrete topology.

(a)

Explain why the set of all “horizontal intervals” of the form
\begin{equation*} I= (a,b) \times \{c\} = \{(x,c) \mid a \lt x \lt b\} \end{equation*}
is a base for the product topology on \(X \times Y\text{.}\)

(b)

Find the interior and closure of each of the following subsets of \(X \times Y\text{.}\)
(i)
\(A = \{(x,0) \mid 0 \leq x \lt 1\}\)
(ii)
\(B = \{(0,y) \mid 0 \leq y \lt 1\}\)
(iii)
\(C = \{(x,y) \mid 0 \leq x \lt 1, 0 \leq y \lt 1\}\)

3.

Let \(X_1\) and \(X_2\) be topological space and let \(A_1\) be a subset of \(X_1\) and \(A_2\) a subset of \(X_2\text{.}\) Assume the product topology on \(X_1 \times X_2\text{.}\) Prove each of the following.

(a)

\(\overline{A_1 \times A_2} = \overline{A_1} \times \overline{A_2}\)

(b)

\(\Int(A_1 \times A_2) \subseteq \Int{A_1} \times \Int{A_2}\)

4.

Let \(X\) and \(Y\) be topological spaces and let \(\{U_{\alpha}\}_{\alpha \in I}\) and \(\{V_{\beta}\}_{\beta \in J}\) be collections of open sets in \(X\) and \(Y\text{,}\) respectively, for some indexing sets \(I\) and \(J\text{.}\) Show that
\begin{equation*} \bigcup_{\substack{\alpha \in I \\ \beta \in J} } (U_{\alpha} \times V_{\beta}) = \left(\bigcup_{\alpha \in I} U_{\alpha} \right) \times \left(\bigcup_{\beta \in J} V_{\beta} \right)\text{.} \end{equation*}

5.

(a)

If \(S_1\text{,}\) \(S_2\text{,}\) \(T_1\text{,}\) and \(T_2\) are sets, show that
\begin{equation*} (S_1 \times T_1) \cap (S_2 \times T_2) = (S_1 \cap S_2) \times (T_1 \cap T_2)\text{.} \end{equation*}

(b)

If \(\CB_X\) is a base for a topology \(\tau_X\) on a space \(X\) and \(\CB_Y\) is a base for a topology \(\tau_Y\) on a space \(Y\text{,}\) show that \(\CB_X \times \CB_Y\) is a base for the product topology on \(X \times Y\text{.}\)

6.

Prove that the product of any finite number of connected spaces is connected.

7.

Prove that the product of any finite number of compact spaces is compact.

8.

(a)

Prove that if \((X_1, \tau_1)\) and \((X_2, \tau_2)\) are path connected topological spaces, then \(X_1 \times X_2\) with the product topology is a path connected topological space.

(b)

Prove that the product of any finite number of path connected spaces is path connected.

9.

Let \(X_1\) and \(X_2\) be topological spaces and \(X = X_1 \times X_2\text{.}\) Is it true that if \(X\) is compact, then both \(X_1\) and \(X_2\) are compact. Prove your answer.

10.

Let \(X_1\) and \(X_2\) be topological spaces and let \(\pi_i : X_1 \times X_2 \to X_i\) be the projection mapping. We have shown that \(\pi_i\) is continuous. Now show that \(\pi_i\) is an open map for \(i\) equal 1 and 2. Assume the standard product topology.

11.

Let \(X_1\) and \(X_2\) be topological spaces with cardinalities at least 2. Let \(X = X_1 \times X_2\text{.}\) Prove that the product space topology on \(X = X_1 \times X_2\) is the discrete topology if and only if the topologies on \(X_1\) and \(X_2\) are the discrete topologies.

12.

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. Throughout, let \(X_1\) and \(X_2\) be topological spaces and \(X = X_1 \times X_2\) with the product topology.

(a)

If \(A_1 \subseteq X_1\) and \(A_2 \subseteq X_2\text{,}\) then \((X_1 \times X_2) \setminus (A_1 \times A_2) = (X_1 \setminus A_1) \times (X_2 \setminus A_2)\text{.}\)

(b)

If \(A_1 \subseteq X_1\) and \(A_2 \subseteq X_2\text{,}\) then \(\Bdry(A_1 \times A_2) \subseteq \Bdry{A_1} \times \Bdry{A_2}\text{.}\)

(c)

If \(A_1 \subseteq X_1\) and \(A_2 \subseteq X_2\text{,}\) then \(\Bdry{A_1} \times \Bdry{A_2} \subseteq \Bdry(A_1 \times A_2)\text{.}\)

(d)

If \(O_1\) is an open subset of \(X_1\) and \(O_2\) is an open subset of \(X_2\text{,}\) then \(O_1 \times O_2\) is an open subset of \(X_1 \times X_2\text{.}\)

(e)

If \(O_1\) is a subset of \(X_1\) and \(O_2\) is a subset of \(X_2\) and \(O_1 \times O_2\) is an open subset of \(X_1 \times X_2\text{,}\) then \(O_1\) is an open subset of \(X_1\) and \(O_2\) is an open subset of \(X_2\text{.}\)

(f)

If \(C_1\) is a closed subset of \(X_1\) and \(C_2\) is a closed subset of \(X_2\text{,}\) then \(C_1 \times C_2\) is a closed subset of \(X_1 \times X_2\text{.}\)

(g)

If \(C_1\) is a subset of \(X_1\) and \(C_2\) is a subset of \(X_2\) and \(C_1 \times C_2\) is a closed subset of \(X_1 \times X_2\text{,}\) then \(C_1\) is a closed subset of \(X_1\) and \(C_2\) is a closed subset of \(X_2\text{.}\)
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