1.
Determine exactly which finite topological spaces are Hausdorff. Prove your result.
Exercises Exercises
2.
Let \((X, \tau)\) be a topological space and let \(A\) be a subset of \(X\text{.}\) Prove that \(\overline{A} = A \cup \Bdry(A)\text{.}\)
3.
Let \(A\) a subset of a topological space. Prove that \(\Bdry(A) = \emptyset\) if and only if \(A\) is open and closed.
4.
Let \(X\) be a nonempty set with at least two elements 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. Let \(A = (0,1]\) be a subset of \(\R\text{.}\) Find, with proof, \(\overline{A}\text{,}\) \(\Int(A)\text{,}\) and \(\Bdry(A)\) when
(a)
\(\R\) has the topology \(\tau_{p}\) with \(p = 0\)
(b)
\(\R\) has the topology \(\tau_{\overline{p}}\) with \(p = 0\text{.}\)
5.
Let \(\B = \{[a,b) \mid a \lt b \text{ in } \R\}\text{.}\)
(a)
Show that \(\B\) is a basis for a topology \(\tau_{\ell \ell}\) on \(\R\text{.}\) This topology is called the lower limit topology on \(\R\text{.}\) The line \(\R\) with the topology \(\tau_{\ell \ell}\) is sometimes called the Sorgenfrey line (after the mathematician Robert Sorgenfrey).
(b)
Show that every open interval \((a,b)\) is also an open set in the lower limit topology.
(c)
If \(\tau_1\) and \(\tau_2\) are topologies on a set \(X\) such that \(\tau_1 \subseteq \tau_2\text{,}\) then \(\tau_1\) is said to be a coarser topology that \(\tau_2\text{,}\) or \(\tau_2\) is a finer topology that \(\tau_1\text{.}\) Part (b) shows that the lower limit topology may be a finer topology than the Euclidean metric topology. Determine if this is true, that the lower limit topology is actually a finer topology than the Euclidean metric topology on \(\R\text{.}\) Justify your answer.
(d)
Let \(a \lt b\) be in \(\R\text{.}\) Is the set \([a,b)\) clopen in \((\R, \tau_{\ell \ell})\text{?}\) Prove your answer.
6.
A subset \(A\) of a topological space \(X\) is said to be dense in \(X\) if \(\overline{A} = X\text{.}\)
(a)
Show that \(\Q\) is dense in \(\R\) using the Euclidean metric topology.
(b)
Is \(\Z\) dense in \(\R\) using the Euclidean metric topology? Prove your answer.
(c)
Let \(A\) be a subset of a topological space \(A\text{.}\) Prove that \(A\) is dense in \(X\) if and only if \(A \cap O \neq \emptyset\) for every open set \(O\text{.}\)
7.
Let \(X\) be a topological space and let \(A\) be a subset of \(X\text{.}\)
(a)
Show that the sets \(\Int(A)\text{,}\) \(\Bdry(A)\text{,}\) and \(\Int(A^c)\) are mutually disjoint (that is, the intersection of any two of these sets is empty).
(b)
Prove that \(X = \Int(A) \cup \Bdry(A) \cup \Int(A^c)\text{.}\)
8.
Prove that a subspace of a Hausdorff space is a Hausdorff space.
9.
Let \(X\) be a nonempty set with at least two elements 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, if \(X\) is a Hausdorff space when
(a)
\(X\) has the topology \(\tau_{p}\)
(b)
\(X\) has the topology \(\tau_{\overline{p}}\text{.}\)
10.
Prove that a subset \(C\) of a topological space \(X\) is closed if and only if \(C\) contains its boundary.
11.
Recall that a point \(a\) in a subset \(A\) of a metric space \(X\) is an isolated point of \(A\) if there is a neighborhood \(N\) of \(a\) in \(X\) such that \(N \cap A = \{a\}\text{.}\) We can make the same definition in any topological space.
Definition 13.17.
A point \(a\) in a subset \(A\) of a topological space \(X\) is an isolated point of \(A\) if there is a neighborhood \(N\) of \(a\) such that \(N \cap A = \{a\}\text{.}\)
(a)
If \(A\) is a subset of a topological space \(X\text{,}\) prove that a point \(a \in A\) is an isolated point of \(A\) if and only if \(\{a\}\) is an open set in \(A\text{.}\)
(b)
We proved that in a metric space every boundary point of a set \(A\) is either a limit point or an isolated point of \(A\text{.}\) (See Exercise 12.) Is the same statement true in a topological space? Prove your answer.
12.
For each integer \(a\text{,}\) let \(a\Z = \{ka \mid k \in \Z\}\text{.}\) That is, \(a\Z\) is the set of all integer multiples of \(a\text{.}\) That \(\{a\Z \mid a \in \Z\}\) is a basis for a topology \(\tau\) on \(\Z\) is the topic of Exercise (ex_aZ_top). In this exercise work in the topological space \((\Z, \tau)\)
(a)
Let \(A = \mathbb{E}\text{,}\) the set of even integers.
(i)
Find, with justification, \(\Int(A)\text{.}\)
(ii)
Find, with justification, \(\overline{A}\text{.}\)
(b)
Let \(B = \mathbb{N} = \{n \in \Z \mid n \geq 1\}\text{.}\) That is, \(\mathbb{N}\) is the set of natural numbers.
(i)
Find, with justification, \(\Int(B)\text{.}\)
(ii)
Find, with justification, \(\overline{B}\text{.}\)
13.
Consider the Double Origin topology defined as follows. Let \(X = \R^2 \cup \{0^*\}\text{,}\) where \(0^*\) is considered as a point that is not in \(\R^2\) (\(0^*\) is our double origin). As a basis for the open sets, we use the standard open balls for every point except \(0\) and \(0^*\text{.}\) For the point \(0\text{,}\) we define open sets to be
\begin{equation*}
N(0,r) = \left\{(x,y) \in \R^2 \mid x^2+y^2 \lt \frac{1}{r^2}, y \gt 0\right\} \cup \{0\}
\end{equation*}
and for \(0^*\) we define open sets to be
\begin{equation*}
N(0^*, r) = \left\{(x,y) \in \R^2 \mid x^2+y^2 \lt \frac{1}{r^2}, y \lt 0\right\} \cup \{0^*\}\text{.}
\end{equation*}
So \(N(0,r)\) is the top half of a disk of radius \(\frac{1}{r}\) centered at the origin, excluding the \(y\)-axis but including the origin, and \(N(0^*,r)\) is the bottom half of a disk of radius \(\frac{1}{r}\) centered at the origin, excluding the \(y\)-axis and including the point \(0^*\text{.}\)
(a)
Show that the collection of sets described as a basis for the Double Origin topology is actually a basis for a topology.
(b)
Is \(X\) with the Double Origin topology Hausdorff? Prove your answer.
14.
(a)
Show that finite sets are closed in \(\R^n\) with the Zariski topology.
(b)
Show that \(\R^n\) with the Zariski topology is not Hausdorff. (Exercise 12 shows that a basis for the Zariski topology on \(\R^n\) is the collection of sets of the form \(\R^n \setminus Z(f)\text{,}\) where \(Z(f)\) is the set of zeros of the polynomial \(f\) in \(n\) variables.)
15.
Consider the digital line topology \(\tau_{dl}\) on \(\Z\) with basis \(\{B(n)\}\text{,}\) where
\begin{equation*}
B(n) = \begin{cases}\{n\} \amp \text{ if \(n\) is an odd integer } , \\ \{n-1,n,n+1\} \amp \text{ if \(n\) is an even integer } . \end{cases}
\end{equation*}
(a)
Let \(A = \{-1,0,1\}\) of \((\Z, \tau_{dl})\text{.}\)
(i)
Find the limit points and boundary points of \(A\text{.}\) Prove your conjectures. Is every limit point of \(A\) a boundary point of \(A\text{?}\) Is every boundary point of \(A\) a limit point of \(A\text{?}\)
(ii)
Find \(\overline{A}\) and write \(X \setminus \overline{A}\) as a union of open sets.
(b)
Now consider the subset \(B = \{0\}\) of \((\Z, \tau_{dl})\text{.}\)
(i)
Find the limit points and boundary points of \(B\text{.}\) Prove your conjectures. Is every limit point of \(B\) a boundary point of \(B\text{?}\) Is every boundary point of \(B\) a limit point of \(B\text{?}\)
(ii)
Find \(\overline{B}\) and write \(X \setminus \overline{B}\) as a union of open sets.
16.
Let \((X,d)\) be a metric space. Recall from Exercise 8 in Chapter 10, that if \(C\) is a closed subset of a metric space \(X\) and \(x\) is an element of \(X \setminus C\text{,}\) then \(d(x,C) = 0\) if and only if \(x \in C\text{.}\) Use this idea to do the following.
(a)
Prove that every metric space is regular.
(b)
Prove that every metric space is a normal space.
17.
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. That \(\B\) generates a topology \(\tau_K\) on \(\R\) follows from the fact that \(\tau_K\) is finer than the Euclidean topology.
(a)
Show that \((\R, \tau_K)\) is a Hausdorff space.
(b)
Exercise 16 shows that every metric space is regular. In this part of the exercise, show that \((R, \tau_K)\) is not a regular space. We can conclude that \((\R, \tau_K)\) is not metrizable.
Hint.
Consider \(0\) and \(K\text{.}\)
18.
(a)
Prove that a topological space \(X\) is \(T_1\) if and only if each singleton set is closed.
(b)
Show that every \(T_2\)-space is \(T_1\text{,}\) that every \(T_3\)-space is \(T_2\text{,}\) and that every \(T_4\)-space is \(T_3\text{.}\)
19.
In this exercise we illustrate spaces that are \(T_1\) but not \(T_2\) and \(T_2\) but not \(T_3\text{.}\)
(a)
Show that \(\R\) with the finite complement topology is \(T_1\) but not \(T_2\text{.}\)
(b)
Define the space \(\R_K\) as in Example 13.13 to be the set of reals with topology \(\tau\) with a basis that consists of the standard open intervals in \(\R\) along with all sets of the form \((a,b) \setminus K\text{,}\) where \((a,b)\) is any open interval and \(K = \left\{\frac{1}{k} \mid k \in \Z^+\right\}\text{.}\) Show that \(\R_K\) is \(T_2\) but not \(T_3\text{.}\)
20.
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)
Every limit point of a subset \(A\) of a topological space \(X\) is also a boundary point of \(A\text{.}\)
(b)
Every boundary point of a subset \(A\) of a topological space \(X\) is also a limit point of \(A\text{.}\)
(c)
If \(X\) is a topological space and \(A \subseteq X\) such that \(\Int(A)=\overline{A}\text{,}\) then \(A\) is both open and closed.
(d)
If \(X\) is a topological space and \(A\) and \(B\) are subsets of \(X\) with \(\overline{A}=\overline{B}\) and \(\Int(A) = \Int(B)\text{,}\) then \(A = B\text{.}\)
(e)
If \(A\) and \(B\) are subsets of a topological space \(X\text{,}\) then \(\overline{A \cap B} = \overline{A} \cap \overline{B}\text{.}\)
(f)
If \(A\) and \(B\) are subsets of a topological space \(X\text{,}\) then \(\overline{A \cup B} = \overline{A} \cup \overline{B}\text{.}\)
