Page 39 - DMTH503_TOPOLOGY

Unit 2: Basis for Topology

Notes
Example 4: Find out a sub-base  for the discrete topology T on X = {a, b, c} s.t.  does not
contain any singleton set.
Solution: Let X = {a, b, c}. Let T be the discrete topology on X.
If we write  = {{x} : x  X}, then by the theorem:
“Let X be an arbitrary set and  a non empty subset of the power set P(X) of X.  is a base for some
topology on X iff
(i) {B : B  } = X
(ii) x  B , B and B , B ,     B   s.t. x  B  B  B .
1 2 1 2 1 2
 is a base for the topology T on X.”
Any family B* of subsets of X.  does not contain any singleton set. Hence,  is the required
sub-base.

Self Assessment

5. Let  be a sub-base for the topologies T and T on X. Show that T = T .
1 1
6. Let (Y, ) be a sub-base of (X, T) and  a sub-base for T on X. Show that the family {Y  S :
S  } is a sub-base for on Y.

7. Given a non empty family  of subsets of a set X, show that  weakest topology T on X in
which all the members of  are open sets and  is a sub-base for T.
8. Let X = {a, b, c, d, e}. Find a sub-base  for the discrete topology T on X which does not
contain any singleton set.

2.3 Standard Topology and Lower Limit Topology

2.3.1 Standard Topology

If  = {(a, b) : a, b  R s.t. a < b} i.e.  is a collection of open intervals on real line, the topology
generated by  is called standard topology on .

2.3.2 Lower Limit Topology

If  = {(a, b] : a, b  R and a < b} i.e.  is a collection of semi-open intervals, the topology
1 1
generated by  is called lower limit topology on .
1
When  is given the lower limit topology, we denote it by  .
1
Finally let K denote the set of all numbers of the form , for n  Z and let  be the collection
n
2
+
of all open intervals (a, b) along with all sets of the form (a, b) –K. The topology generated by 
2
will be called the K-topology on . When  is given this topology, we denote it by  .
K
Lemma: The topologies of  and  are strictly finer than the standard topology on , but are
l k
not comparable with one another.
Proof: Let T, T and T be the topologies or  ,  and  , respectively. Given a basis elements
l k
(a, b) for T and a point x of (a, b), the basis element [x, b) for T contains x and lies in (a, b). On the
other hand, given the basis element [x, b) for T, there is no open interval (a, b) that contains x and
lies in [x, d). Thus T is strictly finer than T.

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