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Definition: Open Sets in Metric Spaces

Let $(X,d)$ be a metric space. A subset $U\subseteq X$ is open, if for each $x\in U$ there exists $\epsilon > 0$ such that the open ball is contained in $U$: $$B(x,\epsilon)\subset U,$$ i.e. if $U$ it is the neighborhood of any of its points. Please note that, by definition, every open ball $B(x, r)$ is open.

Table of Contents

1. Lemma: Characterization of Closed Sets by Limits of Sequences
2. Definition: Open Function, Closed Function

Mentioned in:

Definitions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Lemmas: 15
Parts: 16 17
Proofs: 18 19 20 21 22 23 24 25 26
Propositions: 27 28 29
Theorems: 30

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References

Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im $\mathbb R^n$, Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984