Let S Connected subset of R^2 and p $\displaystyle \in$ R^2 accumulation point. Show that {p} union S is conected. What happends if p isnt a acummulation point?

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- Jun 6th 2008, 12:59 PMkezmanOpen set
Let S Connected subset of R^2 and p $\displaystyle \in$ R^2 accumulation point. Show that {p} union S is conected. What happends if p isnt a acummulation point?

- Jun 6th 2008, 02:46 PMPlato
Proofs on connected sets depend on the sequence of text material you have.

Look for a theorem about the existence of two disjoint open sets that separate $\displaystyle \{ p \}\cup S$.

Note any open set that contains $\displaystyle p$ must contain another point of $\displaystyle S$.

But would that separate $\displaystyle S$?