Recently, I came upon an interesting topology*, the VIP topology. By interesting, I mean that it is useful for producing some pretty cool counterexamples. 

It is defined as follows:
X is any (non-empty) set. a is a fixed element of X. The subset U of X is open if either it is empty, or if it contains a.

It is easy to verify that the collection T(X) of such open sets defined above is a Topology on X. (a is called the VIP for obvious reasons). The topology, called VIP topology satisfies the following properties:

  •  There always exists a basis of T(X) with cardinality equal to X. Just take the basis to be B(X) = { {a,x} : x is an element of X }. Note that this topology is different from the Discrete Topology, which is generated by singletons as basic open sets. 

  • The VIP topology is T0 (i.e. Hausdorff) but not T½. Given any two distinct points x and a,there exists the open set {a} containing a but not x. A symmetric argument does not work if x and a are interchanged, which means that VIP is not T½.   

  • VIP is connected. For, to exist separation into two disjoint open sets, each must contain a or one of them must be empty.

A similar topology is obtained if one replaces the necessity of ‘inclusion’ of a to ‘exclusion’. In other words, a subset U of X is open if it does not contain the (fixed) element a of X (unless U is the whole space X, just to avoid the triviality trap!) Such a topology (check!) is called the Outcast Topology. Just like the VIP, it has some interesting properties, one of them being that it is trivially a compact space!

* Reference: 
The ordered pair (X, T(X)) is a topological space if T(X) is a subset of the power-set of X containing the empty set, the whole set X along with arbitrary unions and finite intersections of elements of T(X).

A (topological) space is T0, if for any two distinct points, there exists an open set containing one but not the other. 
A space is if for two distinct points x and y, there exists an open set containing x but not y.
A space is T1 if given two distinct points x and y, there exist disjoint open sets U and V containing x and y respectively. This is the same as Hausdorff space.

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