Definition 4.1.21

A mapping  is called bicontinuous or topological  if is open and continuous.

It is clear that,

homoeomorphism=bicontinuous+bijection.

 

Definition 4.1.22

  A property  of sets is called topological or a topological invariant if whenever a topological space  has  then every space homoeomorphic to  has also .

 

Example 4.1.23

  As seen before the real line  is homoeomorphic to the open interval . Hence the length is not a topological property.

 

Example 4.1.24

  A topological space is discon­nected if  is the union of two non-empty open disjoint sets, i.e., , where, . If is a homoeomorphism, then  if  and so  is disconnected if   is disconnected.

 

4.2 - PREOREN SETS AND PRECONTINUITY

Definition 4.2.1

  In a space  a subset  is called preopen if . The collection of all preopen sets in  is denoted by .

 

Theorem 4.2.2

The union of all preopen sets of a space  is preopen.

 

Proof

Let  , then by Definition 4.2.1., Thus, , for every .

 

Definition 4.2.3

Let  be a non empty set, then a class of subsets of  is called a supratopology on  if

The union of members of  is also a member of  . It is clear that  forms a supratopology on the space .

 

Remark 4.2.4

The intersection of a finite number of preopen sets of a topological space need not be preopen, as is illustrated in the following example, and therefore  may fail to be a topology on .

 

Example 4.2.5

 Let  with topology .

Then , .

      Observe that .

      It is clear that every open set is preopen, but the converse may not be true, is preopen in Example 4.2.5 but not open.

 

Example 4.2.6

  For an indiscrete space , containing more than one point,   every subset is preopen since  for every .

 

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