Find a topological space \(X\) and a point \(x \in X\) such that the minimal neighborhood of \(x\) is not an open set.
2.
Let \(X\) be a topological space and for each \(x \in X\) let \(PC(x)\) denote the path component of \(x\text{.}\) Prove the following.
(a)
If \(A\) is a path connected subset of \(X\text{,}\) then \(A \subseteq PC(x)\) for some \(x \in X\text{.}\)
(b)
The space \(X\) is path connected if and only if \(X = PC(x)\) for some \(x \in X\text{.}\)
3.
In Activity 18.7 of Chapter 18 we showed that an arbitrary union of connected sets is connected provided the intersection of those sets is not empty. Is the same result true for path connected sets. That is, if \(X\) is a topological space and \(\{A_{\alpha}\}\) for \(\alpha\) in some indexing set \(I\) is a collection of path connected subsets of \(X\) and \(\bigcap_{\alpha \in I} A_{\alpha} \neq \emptyset\text{,}\) must it be the case that \(A = \bigcup_{\alpha \in I} A_{\alpha}\) is path connected? Prove your answer.
4.
Let \(X\) be the subspace of \((\R^2, d_E)\) consisting of the line segments joining the point \((0,1)\) to every point in the set \(\left\{\left(\frac{1}{n}, 0\right) \mid n \in \Z^+\right\}\) as illustrated in Figure 19.19. This space is called the harmonic broom.
Figure19.19.The harmonic broom.
(a)
Show that the harmonic broom is connected.
(b)
Show that the harmonic broom is path connected.
(c)
Show that the harmonic broom is not locally connected.
(d)
Show that the harmonic broom is not locally path connected. So path connectedness does not imply local path connectedness.
5.
In Exercise 4 we see an example of a space that is path connected but not locally path connected. Is it possible to find a space that is locally path connected but not path connected? Verify your answer.
6.
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 not path connected. (Hint: Suppose there is a path between \(a\) and \(b\) where \(a \lt 0\) and \(b \gt 1\text{.}\))
7.
We know that a space can be connected but not path connected. We also know that local path connectedness does not imply connectedness. However, if we combine these conditions then a space must be path connected. That is, show that if a topological space \(X\) is connected and locally path connected, then \(X\) is path connected.
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. Determine, with proof, the path connected subsets of \(X\) when
(a)
\(X\) has the particular point topology \(\tau_p\)
(b)
\(X\) has the excluded point topology \(\tau_{\overline{p}}\text{.}\)
9.
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 \(X\) is a path connected topological space, then any subspace of \(X\) is path connected.
(b)
If \(A\) and \(B\) are path connected subspaces of a topological space \(X\text{,}\) then \(A \cap B\) is path connected.
(c)
There is no path from \(a\) to \(b\) in \((X, \tau)\text{,}\) where \(\tau\) is the discrete topology.
(d)
If \(X\) is a compact locally path connected topological space, then \(X\) has only finitely many path components.
(e)
Every locally path connected space is locally connected.
