1.
(a)
Determine the compact subsets of a topological space \(X\) with the indiscrete topology.
(b)
Determine the compact subsets of a topological space \(X\) with the indiscrete topology.
Exercises Exercises
2.
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 compactness under \(\tau_1\) or \(\tau_2\) imply, if anything, about compactness under the other topology? Justify your answers.
3.
Let \(\E\) be the set of even integers, and let \(\tau = \{\Z\} \cup \{O \subseteq \E\}\text{.}\) That is, \(\tau\) is the collection of all subsets of \(\E\) along with \(\Z\text{.}\)
(a)
Prove that \(\tau\) is a topology on \(\Z\text{.}\)
(b)
Find all compact subsets of \((\Z, \tau)\text{.}\) Verify your answer.
(c)
Prove or disprove: If \(A\) and \(B\) are compact subsets of a topological space \(X\text{,}\) then \(A \cap B\) is also a compact subset of \(X\text{.}\)
4.
Let \((X,\tau)\) be a topological space
(a)
Prove that the union of any finite number of compact subsets of \(X\) is a compact subset of \(X\text{.}\)
(b)
In Exercise 3 we should have seen that the intersection of compact sets is not necessarily compact. If \(X\) is Hausdorff, prove that the intersection of any finite number of compact subsets of \(X\) is a compact subset of \(X\text{.}\)
5.
Consider \(\Z\) with the digital line topology (see Exercise 11). Determine the compact subsets of \(\Z\text{.}\)
6.
For each \(n \in \Z^+\text{,}\) let \((-n,n)\) be the set of integers in the interval \((-n,n)\) (see Exercise 4.)
(a)
Show that \(\B = \{(-n,n)\}_{n \in \Z^+}\) is a basis for a topology \(\tau\) on \(\Z\)
(b)
Is the subset \((-2,2)\) compact in this topology?
(c)
Determine all of the compact subsets of \(\Z\text{.}\)
7.
Let \(X\) and \(Y\) be topological spaces, and let \(f: X \to Y\) be a function.
(a)
Suppose that \(f\) is a continuous function, and that \(X\) is compact and \(Y\) is Hausdorff. Prove that if \(C\) is a closed subset of \(X\text{,}\) then \(f(C)\) is a closed subset of \(Y\text{.}\) (Thus, \(f\) is a closed function.)
Hint.
Use Activity 17.3, Activity 17.4, and Theorem 17.6.
(b)
Suppose that \(f\) is a continuous bijection. Prove that if \(X\) is compact and \(Y\) is Hausdorff, then \(f\) is a homeomorphism.
(c)
Give an example where \(f\) is a continuous bijection and \(X\) is compact, but \(f\) is not a homeomorphism.
(d)
Give an example where \(f\) is a continuous bijection and \(Y\) is Hausdorff, but \(f\) is not a homeomorphism.
8.
The Either-Or topology on the interval \(X = [-1,1]\) has as its open sets all subsets of \(X\) that contain \((-1,1)\) and any subset of \(X\) that doesn’t contain \(0\text{.}\)
(a)
Describe the non-trivial closed subsets of \(X\text{.}\)
(b)
Is \(X\) a Hausdorff topological space? Explain.
(c)
Is \(X\) compact? Prove your answer.
(d)
Are there any subsets of \(\Z\) that are not compact? Justify your answer.
9.
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 \(\B\text{.}\)
(a)
Show that \((\R, \tau_K)\) is not compact.
Hint.
How is the \(K\)-topology related to the Euclidean topology?
(b)
Show that any subset of \(\R\) that contains \(K\) is not a compact subset of \((\R, \tau_K)\text{.}\) In particular, even though \([0,1]\) is a closed and bounded subset of \(\R\) in \((\R, \tau_K)\text{,}\) we note that \([0,1]\) is not compact.
Hint.
Consider the sets \(O_k = \left(\frac{1}{k},2\right) \cup (-1,1) \setminus K\) for \(k \in \Z^+\text{.}\)
10.
Let \(X\) be a topological space.
(a)
Prove that if \(X\) is Hausdorff and \(C\) is a compact subset of \(X\text{,}\) then for each \(x \in X \setminus C\) there exist disjoint open sets \(U\) and \(V\) such that \(x \in U\) and \(C \subseteq V\text{.}\)
(b)
Prove that if \(X\) is a compact Hausdorff space, then \(X\) is normal.
11.
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 compact subsets of \(X\) when
(a)
\(X\) has the particular point topology \(\tau_p\)
(b)
\(X\) has the excluded point topology \(\tau_{\overline{p}}\text{.}\)
12.
In this exercise we encounter a non-Hausdorff topological space in which single points sets are closed, and in which compact subsets need not be closed. Consider the set \(\Z\) with the finite complement topology \(\tau_{FC}\text{.}\)
(a)
Show that every single point set is closed.
(b)
Explain why \((\Z, \tau_{FC})\) is not a Hausdorff space.
(c)
Let \(U\) be an open set in \((\Z, \tau_{FC})\) that contains \(0\) and \(V\) an open set in \((\Z, \tau_{FC})\) that contains \(1\text{.}\) Explain why it cannot be the case that \(U\) and \(V\) are disjoint — that is, \(U \cap V\) must be non-empty.
(d)
Show that the subset \(\E\) of even integers is a compact subset of \((\Z, \tau_{FC})\) that is not closed. Verify your result.
13.
Let \((X, \tau)\) be a Hausdorff topological space.
(a)
Let \(x \in X\) and let \(A\) be a compact subset of \(X\text{.}\) Prove that there exist disjoint open subsets \(U\) and \(V\) of \(X\) such that \(x \in U\) and \(A \subseteq V\text{.}\)
(b)
Let \(A\) and \(B\) be disjoint compact subsets of \(X\text{.}\) Prove that there exist disjoint open sets \(U\) and \(V\) such that \(A \subseteq U\) and \(B \subseteq V\text{.}\)
14.
Let \((X,\tau_X)\) be a topological space and let \(A\) be a subset of \(X\text{.}\) Let \(\tau_A\) be the subspace topology on \(A\text{.}\) Prove that \(A\) is a compact subset of \(X\) if and only if \((A, \tau_A)\) is a compact topological space.
15.
Let \(X\) be a topological space. A family \(\{F_{\alpha}\}_{\alpha \in I}\) of subsets of \(X\) is said to have the finite intersection property if for each finite subset \(J\) of \(I\text{,}\) \(\bigcap_{\alpha \in J} F_{\alpha} \neq \emptyset\text{.}\) Prove that \(X\) is compact if and only if for each family \(\{F_{\alpha}\}_{\alpha \in I}\) of closed subsets of \(X\) that has the finite intersection property, we have \(\bigcap_{\alpha \in I} F_{\alpha} \neq \emptyset\text{.}\)
16.
Even though \(\R\) is not a compact space, if \(x \in \R\text{,}\) then \(x \in [x-1, x+1]\) and so every point in \(\R\) is contained in a compact subset of \(\R\text{.}\) So if we view \(\R\) from the perspective of a point in \(\R\text{,}\) the space \(\R\) looks compact around that point. This is the idea of local compactness. Locally compact spaces are important in the general topology of function spaces.
Definition 17.15.
A topological space \(X\) is locally compact if for each \(x \in X\) there is an open set \(O\) such that \(p \in O\) and \(\overline{O}\) is compact.
(a)
Explain why \(\R^n\) is locally compact for each \(n \in \Z^+\text{.}\)
(b)
Show that any compact space is locally compact.
(c)
Consider the Sorgenfrey line from Exercise 5. Recall that the Sorgenfrey line is the space \(\R\) with a basis \(\B = \{[a,b) \mid a \lt b \text{ in } \R\}\) for its topology. Show that the Sorgenfrey line is Hausdorff but not locally compact.
17.
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 \(X\) and \(Y\) are compact topological spaces and \(f: X \to Y\) is a continuous bijection, then \(f\) is a homeomorphism.
(b)
If \(X\) is a compact topological space, then any closed subspace of \(X\) is compact.
(c)
If \(X\) is a Hausdorff space, \(Y\) is a compact space, and \(f : X \to Y\) is a continuous and bijective function, then \(f\) is a homeomorphism.
(d)
If \(X\) is a compact space, \(Y\) is a Hausdorff space, and \(f : X \to Y\) is a continuous bijection, then \(f\) is a homeomorphism.
(e)
Let \(C\) be a closed subset of a metric space \((X, d)\) with the metric topology. Then \(C\) is compact.
(f)
If \(A\) is a compact subset of a topological space \(X\text{,}\) then \(A\) is a closed subset of \(X\text{.}\)
(g)
Let \((X,\tau)\) be a topological space with \(\tau\) the discrete topology. Then \(X\) is compact if and only if \(X\) is finite.
