The Excluded Point Topology

The excluded point topology on a set X is a topological structure T that involves excluding a single point p from X.

The collection of subsets of X that are included in the excluded point topology encompasses the following:

• The empty set (Ø)
• The set X itself
• All subsets of X that do not contain the point p

In other words, every open set in the excluded point topology is defined as a subset of X that can be the entire set X, the empty set, or any subset of X provided it does not include the point p.

This definition establishes a topology because it meets the three required criteria to be a topology on an open set.

Note. The uniqueness of this topology lies in its construct around the idea of excluding a specific point, which can lead to interesting and sometimes counterintuitive topological properties.

A Practical Example

Consider a set X consisting of three elements.

$$ X = \{a, b, c\}$$

I choose \(p = a\) as the point to be excluded.

To construct the excluded point topology on X by excluding point p, I must include:

• The empty set Ø
• The entire set X itself, which is X={a, b, c}
• All subsets of X that do not contain the point "a", which in this case are {b}, {c}, {b,c}.

Thus, the excluded point topology on X will be as follows:

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

This collection of sets \(T\) fulfills the properties of a topology:

• The union of any collection of these sets still belongs to T.

  For instance, \(\{b\} \cup \{c\} = \{b, c\}\) and \(\{b\} \cup \emptyset = \{b\}\), both elements of T.

• The intersection of any two of these sets also belongs to T.

  For instance, \(\{b\} \cap \{c\} = \emptyset\) and \(\{b, c\} \cap \{b\} = \{b\}\), both elements of T.

• The empty set \(\emptyset\) and the set X itself are included in T.

This topology demonstrates how, by excluding a specific point (in this case, the element "a"), it is possible to create different "forms" of openness within X.

And so on.