- Objectives
- Continuity
- Definition 4.1.1
- Theorem 4.1.6
- Theorem 4.1.13
- Theorem 4.1.17
- Definition 4.1.21
- Theorem 4.2.7
- Theorem 4.2.11
- Theorem 4.2.18
- Theorem 4.2.24
- Remember
- Tests
- Home

**Definition 4.1.21 **

A mapping _{} is called __bicontinuous__ or __topological __ if _{}is open and continuous.

It is clear that,

**homoeomorphism****=bicontinuous+****bijection****.
**

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 _{}.

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– 7 – 8 – 9
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