Subbasis for a Topology .... Singh, Section 1.4 ....

In summary, the conversation is about understanding a remark by Singh regarding the definition of a sub-basis in topology. The conversation includes an example and a clarification that all finite intersections must also be taken into account when defining the topology.
  • #1
Math Amateur
Gold Member
MHB
3,998
48
TL;DR Summary
I need help in order to understand some remarks by Singh made prior to his definition of a sub-basis ...
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 some remarks by Singh just before he defines a sub-basis ... .. The relevant text reads as follows:
Singh - Start of Sectio 1.4 ... .png

To try to fully understand the above text by Singh I tried to work the following example:
##X = \{ a, b, c \}## and ##\mathcal{S} = \{ \{ a \}, \{ b \} \}##

Topologies containing ##\mathcal{S}## are as follows:

##\mathcal{ T_1 } = \{ X, \emptyset, \{ a, b \} , \{ a, c \}, \{ b, c \}, \{ a \}, \{ b \}, \{ c \} \}##

##\mathcal{ T_2 } = \{ X, \emptyset, \{ a, b \} , \{ a, c \}, \{ a \}, \{ b \} \}##

##\mathcal{ T_3 } = \{ X, \emptyset, \{ a, b \} , \{ b, c \}, \{ a \}, \{ b \} \}##

##\mathcal{ T_4 } = \{ X, \emptyset, \{ a, b \} , \{ a \}, \{ b \} \}##Therefore ##\mathcal{ T } ( \mathcal{S} ) = \mathcal{ T_1 } \cap \mathcal{ T_2 } \cap \mathcal{ T_3 } \cap \mathcal{ T_4 }##

##= \{ X, \emptyset, \{ a, b \} , \{ a \}, \{ b \} \}##
But ... now Singh writes the following ... " ... Clearly ##\mathcal{ T } ( \mathcal{S} )## is the coarsest topology. It consists of ##\emptyset, X##, all finite intersections of members of ##\mathcal{S}## and all unions of these finite intersections. ... ..."

However ... all finite intersections of members of ##\mathcal{S}## comprises ##\{ a \} \cap \{ b \} = \emptyset## ... and so, by this reckoning ... ##\mathcal{ T } ( \mathcal{S} )## consists of ##X## and ##\emptyset## ...
Can someone clarify the above ...

Peter===================================================================================There is a small fragment of relevant text in Singh Section 1.2 ... it reads as follows:
Singh - Propn 1.2.2 ... .png


Hope that helps ... ...

Peter
 
Physics news on Phys.org
  • #2
"Finite intersections" include intersections of only a single set. We can write ##\{a\}\cap\{a\}=\{a\}## so ##\{a\}\in\mathcal{T}(S)##.

Does this clear things up?
 
  • Like
Likes Math Amateur
  • #3
First you take all possible finite intersections, but then you still have to take all possible unions again.

In your example all finite intersections are

##\{a\},\{b\}, \emptyset, X##

Taking all possible unions you get ##\mathcal{T}_4##.
 
  • Like
Likes Math Amateur

Related to Subbasis for a Topology .... Singh, Section 1.4 ....

1. What is a subbasis for a topology?

A subbasis for a topology is a collection of subsets of a given set that can be used to generate the entire topology for that set. It is a smaller set of open sets that can be combined to form all open sets in the topology.

2. How is a subbasis different from a basis?

A subbasis is a smaller set of open sets that can generate the entire topology, while a basis is a larger set of open sets that can generate the entire topology. A subbasis is a subset of a basis, and a basis is a subset of the topology.

3. Can a subbasis generate more than one topology?

Yes, a subbasis can generate more than one topology. This is because different combinations of the subbasis sets can form different topologies.

4. How is a subbasis used to define a topology?

A subbasis is used to define a topology by taking all possible finite intersections of the subbasis sets and their complements. This collection of sets forms a basis for the topology, which can then be used to generate all open sets in the topology.

5. What are some examples of subbasis sets?

Examples of subbasis sets include open intervals on the real line, open rectangles in the plane, and open balls in Euclidean space. In general, any collection of subsets that can be used to generate a topology is considered a subbasis.

Similar threads

I The Order Topology .... .... Singh, Example 1.4.4 .... ....
    • Topology and Analysis
Replies
2
Views
2K
I Local Basis in Topology .... Definitions by Croom and Singh .... ....
    • Topology and Analysis
Replies
14
Views
1K
I Munkres Chapter 5: Problem involving the Tychonoff Theorem
    • Topology and Analysis
Replies
7
Views
2K
MHB Heine-Borel Theorem .... Sohrab, Theorem 4.1.10 .... ....
    • Topology and Analysis
Replies
2
Views
1K
I Convergence in Topological Spaces .... Singh, Example 4.1.1 .... ....
    • Topology and Analysis
Replies
2
Views
2K
I Limit Points & Closure in a Topological Space .... Singh, Theorem 1.3.7
    • Topology and Analysis
Replies
4
Views
2K
MHB Proving Topology in X: A Look at Union & Intersection
    • Topology and Analysis
Replies
2
Views
1K
I Convergence .... Singh, Example 4.1.1 .... .... Another Question ....
    • Topology and Analysis
Replies
2
Views
2K
Definition of a subbasis of a topology
    • Topology and Analysis
Replies
4
Views
2K
MHB Connectedness and Intervals in R .... Another Question .... Stromberg, Theorem 3.47 .... ....
    • Topology and Analysis
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top