Assume that B is a connected subset of X and B intersects both A and X \ A, for some subset A in X. Prove that B intersects the boundary $\displaystyle \partial A = \overline {A}$ \ $\displaystyle int(A)$ where int(A) is the interior of A.

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- Feb 24th 2009, 02:02 PMalakazamBoundaries - interior and exterior
Assume that B is a connected subset of X and B intersects both A and X \ A, for some subset A in X. Prove that B intersects the boundary $\displaystyle \partial A = \overline {A}$ \ $\displaystyle int(A)$ where int(A) is the interior of A.

- Feb 24th 2009, 02:57 PMPlato
Both $\displaystyle Int(A)\;\& \;X\backslash \overline{A}$ are

**disjoint open sets**.

If $\displaystyle \partial (A) \cap B = \emptyset$ that leads at once to a contradiction to B being connected.