Math Infinitum
Mapping the lesson.
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Topology · Lesson 10
Bring the chapter together: check the conditions on a distance, decide what a ball holds, test a set for openness, combine open sets, and separate a neighbourhood from an open set. No worked example sits above the answers.
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Sign in to save progressBring the chapter together: check the conditions on a distance, decide what a ball holds, test a set for openness, combine open sets, and separate a neighbourhood from an open set. No worked example sits above the answers.
Reading a map means knowing which question each symbol answers. These questions differ mainly in which of the chapter's ideas they are asking about.
Decide first what is being asked. A metric question tests three conditions; a ball question compares a distance with a radius; openness asks for a ball around every point; a combination question asks whether the collection is finite; a neighbourhood question is about one point rather than all of them.
: strictly closer than the radius.
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Which rule fails to be a metric on ?
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On the real line, which number lies in ?
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Which subset of the real line is not open?
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Where does come from?
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Which is guaranteed to be open?
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A set carries the discrete metric. Which of its subsets are open in it?
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In , what is between and ?
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Which set is the closed ball on the real line?
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Which sequence converges in with the usual metric?
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A Cauchy sequence in a metric space always converges to a point of that space.
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Which space is complete?
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Which set is open in the space but not in ?
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For and , what is the smallest with for all ?
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Every subset of a metric space is itself a metric space under the same distance rule.