Page 95 - DMTH503_TOPOLOGY

Unit 9: The Metric Topology

Theorem 4: In any metric space, any closed sphere is a closed set. Notes
Proof: Let S 0 r [x ] denote a closed sphere in a metric space (X, d).
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To prove that S 0 r [x ] is a closed set.
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For this we must show that S 0 r [x ] is open in X.
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Let x S  0 r [x ] be arbitrary,
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
x S 0 r [x ]  x  S 0 r [x ]
0
0
 d (x, x ) > r ,  S  y X :d(y,x ) r 0  0  
0 0  0 r [x ] 
0
 d (x, x ) – r > 0
0 0
 r > 0, on taking r = d (x, x ) – r ...(1)
0 0
We claim S r(x)  S 0 r [x ] .
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Let y S be arbitrary, so that, d (y, x) < r.
r (x)
 d (x, x )  d (x, y) + d (y, x ).
0 0
 d (y, x )  d (x, x ) – d (x, y) > d (x, x ) – r = r [on using (1)]
0 0 0 0
 d (y, x ) > r y  S
0 0 0 r [x ]
0
Thus, any y S  y S  0 r [x ]
r (x) 0
 S r(x)  S 0 r [x ] 0


Given any x S  , r 0 s.t. S  S

0 r [x ] r(x) 0 r [x ] 0
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This prove that S 0 r [x ] is open in x.
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Example 3: Give an example to show that the union of an infinite collection of closed sets
in a metric space is not necessarily closed.

Solution: Let 1 ,  1 : n  N be the infinite collection for the usual metric space (, d).
  n
Now each member of this collection is a closed set, being a closed interval.


But  1 ,  1 : n  N    1  1 ,  1   1 3  , 1    0,1 . 

  n  2  
Since ]0, 1] is not closed, it follows that the union of an infinite collection of closed sets is not
closed.

Example 4: Show that every closed interval is a closed set for the usual metric on .

Solution: Let x, y   where x < y. We shall show that [x, y] is closed.
Now  – [x, y] = {a   : a < x or a > y}
= {a   : a < x}  {a   : a > y}
= ]–, x [] y,  [

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