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Unit 2: Basis for Topology

2. Show that the collection  = {(a, b) : a, b  , a < b} of all open intervals in  is a base for Notes
a topology on .
3. Show that the collection  = {[a, b] : a, b  , a < b} of all closed intervals in  is not a base
for a topology on .
4. Show that the collection  = {(a, b] : a, b  , a < b} of half-open intervals is a base for a
topology on .
5. Show that the collection  = {[a, b) : a, b  , a < b} of half-open intervals is a base for a
topology on .

6. Show that if  is a basis for a topology on X, then the topology generated by  equals the
intersection of all topologies on X that contain . Prove the same if  is a sub-basis.
7. If  is a sub-base for the topology T on X, then  = {X, } is also a sub-base for T on X.

Answers: Self Assessment

1.  = {{a, b}, {b, c}, {d}, {b}, , X}

T = {, {a, b, d}, {b, c, d}, {b, d}, {a, b, c}}.
8.  = {{a, b}, {b, c}, {c, d}, {d, e}, {e, a}}.

2.7 Further Readings

Books Engelking, Ryszard (1977), General Topology, PWN, Warsaw.

Willard, Stephen (1970), General Topology, Addison-Wesley. Reprinted 2004, Dover
Publications.

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