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

#### Peter

##### Well-known member
MHB Site Helper
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:

$$\displaystyle X = \{ a, b, c \}$$ and $$\displaystyle \mathcal{S} = \{ \{ a \}, \{ b \} \}$$

Topologies containing $$\displaystyle \mathcal{S}$$ are as follows:

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

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

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

$$\displaystyle \mathcal{ T_4 } = \{ X, \emptyset, \{ a, b \} , \{ a \}, \{ b \} \}$$

Therefore $$\displaystyle \mathcal{ T } ( \mathcal{S} ) = \mathcal{ T_1 } \cap \mathcal{ T_2 } \cap \mathcal{ T_3 } \cap \mathcal{ T_4 }$$

$$\displaystyle = \{ X, \emptyset, \{ a, b \} , \{ a \}, \{ b \} \}$$

But ... now Singh writes the following ...

" ... Clearly $$\displaystyle \mathcal{ T } ( \mathcal{S} )$$ is the coarsest topology. It consists of $$\displaystyle \emptyset, X$$, all finite intersections of members of $$\displaystyle \mathcal{S}$$ and all unions of these finite intersections. ... ..."

However ... all finite intersections of members of $$\displaystyle \mathcal{S}$$ comprises $$\displaystyle \{ a \} \cap \{ b \} = \emptyset$$ ... and so, b this reckoning ... $$\displaystyle \mathcal{ T } ( \mathcal{S} )$$ consists of $$\displaystyle X$$ and $$\displaystyle \emptyset$$ ...

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

Last edited:

#### GJA

##### Well-known member
MHB Math Scholar
Hi Peter ,

$\{a\}\cap\{a\} = \{a\}$ and $\{b\}\cap\{b\}=\{b\}$ are finite intersections of sets contained in $\mathcal{S}$ as well.

#### Peter

##### Well-known member
MHB Site Helper
Hi Peter ,

$\{a\}\cap\{a\} = \{a\}$ and $\{b\}\cap\{b\}=\{b\}$ are finite intersections of sets contained in $\mathcal{S}$ as well.

Thanks GJA ...