
MHB Master
#1
March 24th, 2020,
21:54
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 subbasis ... ..
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

March 24th, 2020 21:54
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