Particular Point Topology

The particular point topology on a set X with a designated point p is defined as the collection of all subsets of X that are either empty or contain the point p.

This topology thus includes the empty set, the entire set X, and any of its subsets that include p.

It is also known as the "fixed point topology".

Note. Being a topology, it must satisfy all the necessary properties of a topology: the inclusion of the empty set and the entire set, as well as the closure under union and intersection operations of subsets.

Example

To construct the particular point topology on X={a,b,c} with the designated point "a", we must include the empty set ∅, the entire set X, and all subsets of X that contain the point "a".

• The empty set: ∅
• The entire set: X={a,b,c}
• All subsets of X that contain "a": {a}, {a,b}, {a,c}

Therefore, the particular point topology for "a" on X is represented by the set:

$$ T=\{ ∅, \{ a \}, \{ a,b \}, \{ a,c \}, \{a,b,c \} \} $$

This collection of sets meets the properties of a topology, as it includes the empty set and the entire set and is closed under union and intersection operations.

• Since every set in T contains the designated point "a", except for the empty set, the union of any collection of these sets will also contain "a", thus being a set that is included in T.
• Considering that all sets in T, except ∅, contain the point "a", the intersection of any finite number of these sets, excluding the intersection with ∅, will contain at least "a".

And so on.