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

[Math Amateur]

TL;DR

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:

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:

Hope that helps ... ...

Peter

 

[Infrared]

"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?

 

[member 587159]

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 $\mathcal{T}_4$.