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Unformatted text preview: Basic Deﬁnitions
Let S ⊂ Rn . and p ∈ Rn .
• S is bounded if it is contained in some
ball in Rn .
• S is a neighborhood of p if S contains
some open ball around P .
• A point p is a limit point of S if every
neighborhood of p contains a point q ∈
S , where q = p .
• If p ∈ S is not a limit point of S , then it
is called an isolated point of S .
• S is closed if every limit point of S is a
point of S .
• A point p ∈ S is an interior point of S if
S contains a neighborhood of p .
• S is open if every point of S is an interior
point of S .
1 • Let S denote all of the limit points of S .
Then the closure S of S is the set S ∪ S .
It is the smallest closed set containing
S and is thus the intersection of all the
closed sets containing S .
• A subset T ⊂ S is dense in S if every
point of S is either in T or a limit point
of T (or both).
R EMARK : These deﬁnitions are unchanged
for any metric space instead of just Rn . 2 ...
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.
- Fall '10