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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:







    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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