Math 440, Fall 2014, Assignment 2

We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

- Voltaire

Carefully define the following terms, then give one example and one non-example of each:

1. Pseudometric.
2. Metric.
3. Continuous at $x=x_0$.
4. $\epsilon$-disc centered at $x_0$.
5. Open set (in a given pseudometric space).
6. Closed set (in a given pseudometric space).
7. Topology (on a set $X$).
8. Stronger (or finer) topology.
9. Weaker (or coarser) topology.
10. Discrete topology.
11. Trivial topology.
12. Open set (in a given topological space).
13. Closed set (in a given topological space).
14. Closure.
15. Interior.
16. Boundary (or frontier).

Carefully state the following theorems (you need not prove them):

1. Fundamental properties of open sets (Theorem 2.6).
2. Theorem characterizing continuity in terms of open sets (Theorem 2.8).
3. Properties of closed sets (Theorem 3.4).
4. Properties of the closure operation (Theorem 3.7).
5. Properties of the interior operation (Theorem 3.11).
6. Theorem relating the boundary of a set to its interior and closure (Theorem 3.14).

Solve the following problems:

1. Consider the metric space $(X,\rho)$, where $X$ is the three-point set $\{a,b,c\}$ and $\rho$ is the discrete metric. If possible, give three distinct sequences that all converge to $a$, and three distinct sequences that do not. At least one of the latter sequences should have no limit at all. If any of these things is not possible, then explain why not.
2. Consider the pseudo-metric space $(X,\rho)$, where $X$ is the three-point set $\{a,b,c\}$ and $\rho$ is the trivial pseudo-metric. If possible, give three distinct sequences that all converge to $a$, and three distinct sequences that do not. At least one of the latter sequences should have no limit at all. If any of these things is not possible, then explain why not.
3. Section 2, problems A, B, and D (for B, recall that "$\mathrm{sup}$" is an abbreviation for "supremum," which in turn is a synonym for "least upper bound;" it is a fundamental property of the real number system that every non-empty set which is bounded above has a unique least upper bound).
4. Section 3, problems A and F(1).

Optional problems (not on the quiz or exams, but recommended):

1. Section 2, problem C. (Part 4 of this problem shows how pseudometrics arise in practice, while parts 1 and 2 show how to turn pseudometric spaces into metric spaces. If you ever study functional analysis, you will find yourself relying heavily on this construction. For example, the $L^p$ spaces begin life as pseudometric spaces, but one quickly replaces them by their metric identifications.)

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Pseudometric:

   Let \(X\) be a set. A pseudometric on \(X\) is a function \(f:X\times X \rightarrow \mathbb{R}_{\geq 0}\) such that:
   \(f(d,d) = 0\) for all \(f\in X\).
   \(f(a,b) = f(b,a)\) for all \(a,b \in X\).
   \(f(a,b) \leq f(a,c) + f(c,b)\) for all \(a,b,c \in X\).

   Example:

   Non-Example:

   If \(f\) maps all pairs to 1, then \(f\) fails to satisfy the first requirement.

2. Metric:

   A metric on \(X\) is a pseudometric on \(X\) with the additional requirement that \(f(a,b) = 0 \rightarrow a=b\) for all \(a,b \in X\).

   Example:

   Normal Euclidean distance on \(\mathbb{R}\) is a metric.

   Non-Example:

3. Continuous at \(x = x_0\):

   Let \(X\) be a metric space. A function \(f:X \rightarrow X\) is continuous at \(x=x_0\) if for all \(\epsilon > 0\) there is a \(\delta > 0\) such that if \(|x - x_0| \lt \delta\), then \(|f(x) - f(x_0)| \lt \epsilon\).

   Example:

   Non-Example:

4. Topology:

   A topology on \(X\) is a set \(T\) of subsets of \(X\), such that:
   \(\emptyset \in T\).
   \(X \in T\).
   For all \(X_i \in T\) such that\(i \in I\), \(\bigcup_{i\in I} X_i \in T\).
   For all \(X_1, X_2 \in T\), \(X_1 \cap X_2 \in T\).

   Example:

   Non-Example:

5. Topology:

   A topology on \(X\) is a set \(T\) of subsets of \(X\), such that:
   \(\emptyset \in T\).
   \(X \in T\).
   For all \(X_i \in T\) such that\(i \in I\), \(\bigcup_{i\in I} X_i \in T\).
   For all \(X_1, X_2 \in T\), \(X_1 \cap X_2 \in T\).

   Example:

   Non-Example: