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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
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



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:


Singh - Propn 1.2.2 ... .png



Hope that helps ... ...

Peter
 
Last edited:

GJA

Well-known member
MHB Math Scholar
Jan 16, 2013
256
Hi Peter ,

$\{a\}\cap\{a\} = \{a\}$ and $\{b\}\cap\{b\}=\{b\}$ are finite intersections of sets contained in $\mathcal{S}$ as well.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
Hi Peter ,

$\{a\}\cap\{a\} = \{a\}$ and $\{b\}\cap\{b\}=\{b\}$ are finite intersections of sets contained in $\mathcal{S}$ as well.


Thanks GJA ...

I appreciate your help ...

Peter