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Oppgaven lyder:
Is every point of every open set [tex]E \subset \mathbb{R}^{2}[/tex] a limit point of [tex]E[/tex]? Answer the same question for closed sets in [tex]\mathbb{R}^{2}[/tex].
Forslag:
Proof by contradiction:
Given the open set [tex]E[/tex]. Suppose there is a point [tex]x \in E[/tex] that is not a limit point of [tex]E[/tex]. Then [tex]x[/tex] is an isolated point of [tex]E[/tex] such that there exists a neighborhood [tex]N_r(x)[/tex] around [tex]x[/tex] with radius [tex]r > 0[/tex] that contains no point other than [tex]x[/tex] in [tex]E[/tex]. However, since we know that [tex]E[/tex] is an open set, every point must be an interior point with a neighborhood [tex]N[/tex] such that [tex]N \subset E[/tex] and where [tex]N[/tex] contains infinitely many points in [tex]E[/tex]. Thus we have reached a contradiction since [tex]x[/tex] contains no such neighborhood. Thus [tex]x[/tex] must be a limit point. The conclusion is that every point must be a limit point.
For del b):
It is not necessary for every point to be a limit point for closed sets in [tex]\mathbb{R}^{2}[/tex]. An example would be the closed set [tex]\{0\}[/tex]. Here the set is closed, but [tex]0[/tex] is not a limit point since it contains no other point in its neighborhood distinct from itself.
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Topologi - spørsmål 2
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Takk! Godt å høre at i hvert fall det er noe jeg får til. Sitter med oppgaver om kompakte sett nå, og føler at jeg ikke får til noen ting! Jeg forstår teoremene og bevisene i boken, men når jeg skal gå løs på egne oppgaver sitter jeg helt fast. Frustrerende!