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- 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

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