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# Thread: Subbasis for a Topology ... Singh, Section 1.4 ... Another Question ... ...

1. 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:

I am unsure of Singh's arguments concerning the nature of $\displaystyle \mathcal{ T } ( \mathcal{S} )$ ... ...

Singh writes the following:

" ... ... Clearly $\displaystyle \mathcal{ T } ( \mathcal{S} )$ is the coarsest topology on X containing $\displaystyle \mathcal{S}$. It consists of $\displaystyle \emptyset, X$, all finite intersections of members of $\displaystyle \mathcal{S}$ and all unions of these finite intersections. This can be easily be ascertained by verifying that the collection of these sets is a topology for X, which contains $\displaystyle \mathcal{S}$ and is coarser than $\displaystyle \mathcal{ T } ( \mathcal{S} )$. ... ... "

My questions are as follows:

Why exactly is the collection of sets specified necessarily coarser than $\displaystyle \mathcal{S}$ ... ?

Indeed ... the comment is confusing since Singh appears to say that the collection of sets mentioned is indeed $\displaystyle \mathcal{ T } ( \mathcal{S} )$ ... ...

Can someone please clarify Singh's argument ...

Help will be much appreciated ...

Peter

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