# Characteristic Function of Metric Space

Let S ⊂ M.

(a) Define the characteristic function Xs : M --> R.

(b) If M is a metric space, show that Xs(x) is discontinuous at x if and only if x is a boundary point of

S.

[Please see attached PDF file for full problem].

for part (a), I think something similar to

http://planetmath.org/encyclopedia/CharacteristicFunction.html

can be used, correct?

https://brainmass.com/math/graphs-and-functions/characteristic-function-metric-space-154837

#### Solution Preview

Please see the attached file for the complete solution.

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Proof:

(a) The characteristic function is defined as

if ; if

(b) Now suppose is a metric space. We should know what is a boundary point of . If is a boundary point of , then in each neighborhood of , we can find some , such that . In another word, any neighborhood of a ...

#### Solution Summary

Characteristic function of a metric space is investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.