What does it mean for a metric space to be complete?
Give two or more equivalently definitions for compactness in a complete metric space.
What is an interior point? An isolated point? A limit point?
What does it mean for a set to be open? Closed?
What is the closure of a subspace \(E\subseteq X\)?
What does it mean for \(E\subseteq X\) to be a dense subspace?
What does it mean for a family of sets to form a basis for a topology?
- What is a basis for the standard topology on \(\RR^d\)?
Let \(X\) be a subset of \(\RR^d\). Prove the Heine-Borel theorem:
Show that \(X\) compact \(\implies X\) is closed
Show that \(X\) compact \(\implies X\) is bounded
Show that a closed subset of a compact set must be bounded.
Show that if \(X\) closed and bounded \(\implies X\) is compact.
Find an example of a metric space with a closed and bounded subspace that is not compact.
- How can this be modified to obtain a necessary and sufficient condition?
Determine if the following subsets of \(\RR\) are opened, closed, both, or neither:
\(\QQ\)
\(\ZZ\)
\(\ts{1}\)
\(\ts{p \in \ZZ^{\geq 0} \st p\text{ is prime}}\)
\(\ts{ {1\over n} \st n\in \ZZ^{\geq 0}}\)
\(\ts{ {1\over n} \st n\in \ZZ^{\geq 0}} \union \ts{0}\)
Solution
Let \((X,d)\) be a metric space and \(E \subseteq X\).
\((X,d)\) is complete if every Cauchy sequence in \(X\) converges to a point of \(X\).
For a metric space \(K\) the following are equivalent: every open cover of \(K\) has a finite subcover (the definition of compactness); every sequence in \(K\) has a subsequence converging in \(K\); \(K\) is complete and totally bounded, that is, for every \(\eps > 0\) finitely many \(\eps\)-balls cover \(K\).
\(x\) is an interior point of \(E\) if \(B(x,r) \subseteq E\) for some \(r > 0\); an isolated point of \(E\) if \(x \in E\) and \(B(x,r) \cap E = \theset{x}\) for some \(r > 0\); a limit point of \(E\) if \((B(x,r)\setminus\theset{x}) \cap E \neq \emptyset\) for every \(r > 0\).
\(E\) is open if every point of \(E\) is an interior point, and closed if \(X \setminus E\) is open, equivalently if \(E\) contains all its limit points.
The closure \(\overline E\) is the smallest closed set containing \(E\), equivalently \(E\) together with its limit points.
\(E\) is dense if \(\overline E = X\), equivalently every nonempty open subset of \(X\) meets \(E\).
A family \(\mathcal B\) of subsets of a set \(X\) is a basis for a topology if every point lies in some \(B \in \mathcal B\), and for \(B_1, B_2 \in \mathcal B\) and \(x \in B_1 \cap B_2\) there is \(B_3 \in \mathcal B\) with \(x \in B_3 \subseteq B_1 \cap B_2\); the topology it generates consists of the unions of members of \(\mathcal B\). The open balls \(B(x,r)\), \(x \in \RR^d\), \(r > 0\), form a basis for the standard topology on \(\RR^d\), as do the open boxes.
1. If \(K \subseteq \RR^d\) is compact, then \(K\) is closed.
1.1. Fix \(x \in \RR^d \setminus K\). For \(y \in K\) the balls \(U_y = B(y, \abs{x-y}/3)\) and \(V_y = B(x, \abs{x-y}/3)\) are disjoint.
\(\abs{x-y} > 0\), and a common point \(z\) would give \(\abs{x-y} \le \abs{x-z} + \abs{z-y} < 2\abs{x-y}/3\).
1.2. There are \(y_1, \ldots, y_k \in K\) with \(K \subseteq U_{y_1} \cup \cdots \cup U_{y_k}\).
The balls \(U_y\), \(y \in K\), form an open cover of the compact set \(K\).
1.3. Q.E.D.
1.1. So every point of \(\RR^d \setminus K\) has a neighborhood disjoint from \(K\), and \(\RR^d \setminus K\) is open.