A mapping is called bicontinuous or topological if is open and continuous.
It is clear that,
A property of sets is called topological or a topological invariant if whenever a topological space has then every space homoeomorphic to has also .
As seen before the real line is homoeomorphic to the open interval . Hence the length is not a topological property.
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.
In a space a subset is called preopen if . The collection of all preopen sets in is denoted by .
The union of all preopen sets of a space is preopen.
Let , then by Definition 4.2.1., Thus, , for every .
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 .
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 .
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.
For an indiscrete space , containing more than one point, every subset is preopen since for every .