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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

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

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