Base or Open Base of a Topology

Let \left( {X,\tau } \right) be a topological space, then the sub collection {\rm B} of \tau is said to be base or bases or open base for \tau , if each member of \tau can be expressed as union of members of {\rm B}.

In other words let \left( {X,\tau } \right) be a topological space, then the sub collection {\rm B} of \tau is said to be base, if for a point x belonging to an open set U then there exist B \in {\rm B} such that x \in B \subseteq U.

Example:

Let X = \left\{ {a,b,c,d,e} \right\} and let \tau  = \left\{ {\phi ,\left\{ {a,b} \right\},\left\{ {c,d} \right\},\left\{ {a,b,c,d} \right\},X} \right\} be a topology defined on X. {\rm B} = \left\{ {\left\{ {a,b} \right\},\left\{ {c,d} \right\},X} \right\} is a sub collection of \tau , which meets the requirement for a base, because each member of \tau is a union of members of {\rm B}.

Remarks:
• It may be noted that there may be more than one base for a given topology defined on that set.
• Since union of empty sub collection of members of {\rm B} is an empty set, so empty set \phi  \in \tau .

For Discrete Topology:

Let X = \left\{ {1,2,3} \right\} and let \tau  = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,3} \right\},X} \right\} be a topology defined on X. {\rm B} = \left\{ {\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\}} \right\} is a base for \tau . Check weather {\rm B} is a base or not take all possible union of {\rm B} there must becomes \tau .

Possible Unions  = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,3} \right\},X} \right\}
In this case discrete topological space, the collection of all singletons subsets of X forms a base for discrete topological space.

For Indiscrete Topology:

Let X = \left\{ {1,2,3} \right\} and let \tau  = \left\{ {\phi ,X} \right\} be a topology defined on X. {\rm B} = \left\{ X \right\} is a base for \tau .

Theorem:
Let \left( {X,\tau } \right) be a topological space, then a sub collection {\rm B}of \tau is a base for \tau if and only if
1. X = \bigcup\limits_{B \in {\rm B}} B
2. If {B_1} and {B_2}belongs to {\rm B}, then {B_1} \cap {B_2} can be written as union of members of {\rm B}. i.e. forx \in {B_1} \cap {B_2} then there exist B of {\rm B} such that x \in B \subseteq {B_1} \cap {B_2}.

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