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- Thread starter Gauss M.D.
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WannabeNewton

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Let ##X## be a topological space and ##A\subseteq X##. We define the **closure** ##\bar{A}## of ##A## as the smallest closed subset of ##X## containing ##A## i.e. ##\bar{A} = \bigcap \left \{ B\subseteq X:B\supseteq A \text{ and B is closed in X} \right \}##. Similarly we define the **interior** ##\text{int}A## as the largest open subset of ##X## contained in ##A## i.e. ##\text{int}A = \bigcup \left \{ C\subseteq X:C\subseteq A \text{ and C is open in X} \right \}##. Using these two topological notions, we define the **boundary** ##\partial A## of ##A## as ##\partial A = X\setminus (\text{intA}\cup (X\setminus \bar{A}))##. Intuitively, it is the part "in between" the exterior and interior of ##A##. For example, the boundary ##\partial A## in ##\mathbb{R}## of the set ## A = (0,1)\subset \mathbb{R}## is just ##\left \{ 0,1 \right \}##, as would be expected.

There is a very nice theorem relating paths in topological spaces to boundaries of sets which makes the concept of the boundary even more geometric but as your professor said these concepts might very well be beyond the scope of your course.

There is a very nice theorem relating paths in topological spaces to boundaries of sets which makes the concept of the boundary even more geometric but as your professor said these concepts might very well be beyond the scope of your course.

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