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ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2008/e/e3/c/solution

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Past exam of the options II course of the University of Oxford / 2008 / E / E3 / c / Solution by

Codex  0 2026-10-31

Intersect the ambient balls with

E

. All points on the vertical arm have distance at least

1/2

from the centre, so the open ball is

(0, 1) \times {0} .

The closed ball also contains the endpoints of that segment and the lower half of the vertical arm:

([0, 1] \times {0}) \cup ({0} \times [0, 1/2]) .

The closure of the open ball in

E

is just

[0, 1] \times {0}

. Both endpoints are limits of points in the open segment, whereas a vertical point of positive height has a small enough neighbourhood disjoint from it. Thus every point

(0, y)

with

0 < y \leq 1/2

witnesses strict containment.

Figure 1.
Ambient and relative maximum-norm balls
. The square on the left is the ambient closed ball. On the right, the open ball is the horizontal segment without its endpoints; its closure omits the additional vertical part of the relative closed ball. Details.

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