# The Order Topology .... .... Singh, Example 1.4.4 .... ....

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In summary, Tej Bahadur Singh provides a definition of the order topology, explains how to generate a basis for the topology, and provides a confirmation that the basis is discrete.
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TL;DR Summary
I need help in order to fully understand the order topology ... so I present a very simple example ...
I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.4: Basis ... ...

I need help in order to fully understand the order topology ... and specifically Example 1.4.4 ... ...Example 1.4.4 reads as follows:

In order to fully understand Example 1.4.4 I decided to take ##X = \{ a, b, c \}## where ##a \leq b, a \leq c## and ##b \leq c## ... ...Now in the above text, Singh writes the following:

" ... ... The basis generated by the subbasis of ##X## consists of all open rays, all open intervals ##(a, b)##, the emptyset ##\emptyset##, and the full space ##X##. ... ... "Now as I understand it the open rays in ##X## are as follows:

##( - \infty, a) = \emptyset##

##( - \infty, b) = \{ a \}##

##( - \infty, c) = \{ a, b \}##

##( a, \infty) = \{ b, c \}##

##( b, \infty) = \{ c \}##

##( c, \infty) = \emptyset##... and (see definition of order topology below) the open rays constitute the subbasis of the order topology ...To generate the basis, according to the text of Example 1.4.4, we have to add in all open intervals ##(a, b)##, the emptyset ##\emptyset##, and the full space ##X##. ... ...

The open intervals in ##X## are as follows:

##(a, b) = \emptyset##

##(b, c) = \emptyset##

##(a, c) = \{ b \}##The above open rays, open intervals together with ##\emptyset## (already in the basis) and ##X## constitute a basis for the order topology of the ordered set ##X## ... ...Can someone please confirm that the above analysis is correct and/or point out errors or shortcomings ... ..
Help will be much appreciated ... ...

Peter======================================================================================It may help Physics Forum readers of the above post to have access to Singh's definitions of the order topology, together with the definitions of subbasis and basis ... so I am providing the same ... as follows:
Hope that helps ... ...

Peter

What you wrote mostly seems fine to me.

- In your definition of ordered set you have to require that ##x\leq x## for all ##x\in X## to make sure the order is reflexive.

- Your above analysis shows that every singelton in your space is open. Thus the order topology here is simply the discrete topology and thus not interesting.

Math Amateur
Thanks for that confirmation...

Gives me confidence...

Peter

member 587159

## 1. What is the Order Topology?

The Order Topology is a mathematical concept used to define a topology on a partially ordered set. It is defined by a collection of open intervals, where an interval is open if it contains all points between any two points in the set. This topology is used to study the properties of ordered sets and is an important tool in the field of topology.

## 2. How is the Order Topology defined?

The Order Topology is defined by a collection of open intervals, where an interval is open if it contains all points between any two points in the set. This means that the open intervals are defined as the sets of points that fall between two points in the ordered set. The union of all these open intervals forms the basis for the Order Topology.

## 3. What are some examples of the Order Topology?

One example of the Order Topology is the real line with the usual ordering. In this case, the open intervals would be the usual open intervals on the real line. Another example is the set of all positive integers with the standard ordering, where the open intervals would be defined as the sets of integers between two given integers.

## 4. How is the Order Topology different from other topologies?

The Order Topology is different from other topologies in that it is specifically designed for partially ordered sets. It takes into account the ordering of the elements in the set and defines open intervals based on this ordering. Other topologies, such as the Euclidean topology, are not dependent on an ordering and can be applied to a wider range of sets.

## 5. What is the significance of Example 1.4.4 in Singh's book?

Example 1.4.4 in Singh's book is significant because it demonstrates the application of the Order Topology to a specific set. In this example, Singh uses the Order Topology to define a topology on the set of all real numbers between 0 and 1, including the endpoints. This example highlights the properties of the Order Topology and how it can be used to study the properties of ordered sets.

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