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Subbasis for a Topology ... Singh, Section 1.4 ... Another Question ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,882
Hobart, Tasmania
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:



Singh - Start of Sectio 1.4 ... .png



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