The following two definitions are taken directly from Rudin's Principles of Mathematical Analysis.(adsbygoogle = window.adsbygoogle || []).push({});

(1) OPEN SUBSET DEFINITION:If [itex]G[/itex] is an open subset of some metric space [itex]X[/itex], then [itex]G \subset X[/itex] and for any [itex]p \in G[/itex] we can find some [itex]r_{p} > 0[/itex] such that the conditions [itex]d(p,q) < r_p[/itex], [itex]q \in X[/itex] implies [itex]q \in G[/itex].

(2) RELATIVE OPENNESS DEFINITION:Suppose [itex]E \subset Y \subset X[/itex]. We say that [itex]E[/itex] is open relative to [itex]Y[/itex] if for each [itex]p \in E[/itex] there is associated an [itex]r_p > 0[/itex] such that [itex]q \in E[/itex] whenever [itex]d(p,q) < r_p[/itex] and [itex]q \in Y[/itex].

1. Couldn't we just say that [itex]E[/itex] is open relative to [itex]Y[/itex] if and only if [itex]E[/itex] is an open subset of [itex]Y[/itex]? Rudin never flat out says this but I just wanted to make sure I wasn't missing something.

2. Suppose [itex]X[/itex] has an isolated point [itex]i[/itex] and it happens that [itex]i \in G[/itex]. While it's true that this would preclude [itex]X[/itex] from being an open set (since an open set must be comprised solely of internal points and [itex]i[/itex] is an isolated point), wouldn't it still be a possibility that [itex]G[/itex] could be an open subset of [itex]X[/itex]? After all, it's (trivially) true that for [itex]i[/itex] we could find some ball [itex]B[/itex] that satisfies the condition set forth in the definition (1) above. If this is true, wouldn't it follow that an open subset is not necessarily itself an open set?

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# Defining Open Subsets in Baby Rudin

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