Consider the seventeen properties listed in Exercise 1 of the Supplementary Exercises (Review of the Basics): connected, path connected, locally connected, locally path connected, compact, limit point compact, locally compact Hausdorff, Hausdorff, regular, completely regular, normal, first-countable, second-countable, Lindelöf, separable, locally metrizable, metrizable.
Which of these properties does a metric space necessarily have?
Solution
Goal: Determine which of the seventeen basic topological properties every metric space \((X, d)\) necessarily satisfies, providing rigorous justifications and counterexamples.
1. Properties that EVERY metric space necessarily satisfies: A metric space \((X, d)\) necessarily has the following 7 properties: 1. Hausdorff (\(T_2\)): For distinct \(x \neq y\), let \(r = \frac{1}{2}d(x, y) > 0\).The open balls \(B(x, r)\) and \(B(y, r)\) are disjoint open neighborhoods.2. Regular (\(T_3\)): Singletons are closed, and for any closed set \(C\) with \(x \notin C\), \(r = d(x, C) = \inf_{c \in C} d(x, c) > 0\).The open sets \(U = B(x, r/2)\) and \(V = \{y \in X \mid d(y, C) < r/2\}\) are disjoint open sets separating \(x\) and \(C\).3. Completely regular (\(T_{3\frac{1}{2}}\)): For any closed set \(C\) and \(x \notin C\), the function \(f(y) = \frac{d(y, x)}{d(y, x) + d(y, C)}\) is continuous with \(f(x) = 0\) and \(f(C) = \{1\}\).4. Normal (\(T_4\)): For disjoint closed sets \(A, B \subseteq X\), the function \(f(y) = \frac{d(y, A)}{d(y, A) + d(y, B)}\) continuously separates \(A\) and \(B\) (or via Stone’s Theorem that metric spaces are paracompact and \(T_1\), hence normal).5. First-countable: For every \(x \in X\), the countable family \(\{B(x, 1/n) \mid n \in \mathbb{Z}_+\}\) forms a countable local neighborhood basis at \(x\).6. Locally metrizable: Every point has an open neighborhood (e.g. \(X\) itself) that is metrizable.7. Metrizable: Holds tautologically since \(X\) is equipped with a compatible metric \(d\).
2. Properties that a metric space does NOT necessarily satisfy (with counterexamples): Proof:
2.1. Connected / Path connected: The discrete two-point metric space \(X = \{0, 1\}\) with \(d(0, 1) = 1\) is neither connected nor path-connected.
2.2. Locally connected / Locally path connected: The subspace \(X = \{0\} \cup \{1/n \mid n \in \mathbb{Z}_+\} \subset \mathbb{R}\) with the Euclidean metric is not locally connected (and not locally path-connected) at \(0\).
2.3. Compact / Limit point compact: The Euclidean line \(\mathbb{R}\) (or \(\mathbb{Z}\) with discrete metric) is neither compact nor limit point compact.
2.4. Locally compact Hausdorff: The rational numbers \(\mathbb{Q} \subset \mathbb{R}\) with the induced Euclidean metric (or the Hilbert space \(\ell^2\)) is not locally compact at any point.
2.5. Second-countable / Lindelöf / Separable: An uncountable set \(X\) equipped with the discrete metric \(d(x, y) = 1\) (\(x \neq y\)) is a metric space that is not second-countable, not Lindelöf, and not separable.
3. Conclusion: The properties possessed by every metric space are precisely: Hausdorff, regular, completely regular, normal, first-countable, locally metrizable, and metrizable. Q.E.D.