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!

**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 *T½* 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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