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Math 131B, Lecture 1
Midterm 1
Fri, April 26, 2024
1. For each of the following, either give an example of a metric space (X, d) that has the given property (and briefly say why), or explain why such a space cannot exist. (Be sure to specify the metric along with the set X!)
(a) (3 points) X has finitely many elements and is not complete.
(b) (3 points) X has a subset E which is closed and whose complement X \ E is also closed.
(c) (3 points) X has a subset E which is closed and bounded and not compact.
2. Consider the metric space (R, dstd ), the real numbers with the standard metric. Let E = (−1, 0] ⊆ R.
(a) (3 points) Show that E can be expressed as the intersection of open subsets of R.
(b) (2 points) Explain why the intersection in part (a) cannot consist of only finitely many open sets.
3. Let (X, d) be a metric space, and suppose x, y ∈ X are distinct points (i.e., x ≠ y).
(a) (3 points) Show that there exists an open set U ⊆ X such that: x ∈ U and y 
U.
(b) (3 points) Show that there exist open sets V, W ⊆ X such that: x ∈ V, y ∈ W, and V ∩ W = ?.