Closed or Open Sets, limit Points and Compactness
A closed set
is a set that includes all it's limit points
This means than for each point
such that there exists a sequence
such that
we must have that![](/university-maths/analysis/compact-open-closed-and-bounded-sets-html-m5edaf338.gif)
![](/university-maths/analysis/compact-open-closed-and-bounded-sets-html-m17d16499.gif)
This means that the boundary of
is in![](/university-maths/analysis/compact-open-closed-and-bounded-sets-html-m11b78cb9.gif)
If the set does not include any of it's boundary then it is said to be open. The interval
is closed, and the interval
is open since the boundary of
is
but
The interval \[(0,1]\]
is neither open or closed since
but![](/university-maths/analysis/compact-open-closed-and-bounded-sets-html-6aa36dd7.gif)
A set
is bounded if and only if there exists
such that for all
This means that the distance however measured from the origin to any element of
is less than or equal to
as shown below.
![](/university-maths/analysis/compact-open-closed-and-bounded-sets-html-bc47d40.gif)
Finally a set is compact if it is both closed and bounded. Compactness has many applications in analysis.
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