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Local Basis in Topology ... Definitions by Croom and Singh ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
Fred H. Croom (Principles of Topology) and Tej Bahadur Singh (Elements of Topology) define local basis (apparently) slightly differently ...

Croom's definition reads as follows:


Croom - Defn of a Local Basis ... .png


... and Singh's definition reads as follows:


Singh - Defn 1.4.10 ... Local basis or nbd basis ... .png


The two definitions appear different ... ...

Croom requires that each open set containing a contains a member of \(\displaystyle \mathcal{B}_a\) ...

while Singh requires each nbd of x to contain some element of a member of \(\displaystyle \mathcal{B}_x\) ... and one notes that the open set (necessarily) contained in the nbd would necessarily intersect with a member of \(\displaystyle \mathcal{B}_x\) ... it would not necessarily contain the member of \(\displaystyle \mathcal{B}_x\) ... so I see the definitions as different ...

Is that correct?

If the definitions are indeed different ... which is the more common definition ...?

Further ... are their significant implications for further theory built on these definitions ...?


Help will be much appreciated ... ...

Peter


=======================================================================================


It may help MHB
readers of the above post to have access to Singh's definitions of a neighborhood (nbd) ...
... so I am providing the same ... as follows:


Singh - Defn 1.2.5 ... ... NBD ... .png


Hope that helps ...

Peter
 

GJA

Well-known member
MHB Math Scholar
Jan 16, 2013
255
Hi Peter ,

Strictly speaking, Singh's definition is slightly more general. However, you can always convert a local basis in the sense of Croom to one in the sense of Singh and vice versa.

Croom to Singh

Since open sets about $a$ are nhoods of $a$, a local basis in the sense of Croom is automatically a local basis in the sense of Singh.

Singh to Croom

Let $\mathcal{B}^{S}_{a}$ be a local basis in the sense of Singh. Each element of $\mathcal{B}^{S}_{a}$ contains an open set about $a$. The collection of all such open sets is a local basis in the sense of Croom.


As you can see, whichever definition you choose is fine. Personally, I prefer the Croom definition because open sets are really what topology "boils down to" in my opinion. I cannot recall a time where working with nhoods provided a distinct advantage to working with open sets. Though it's been awhile since I studied general topology, I recall nhood definitions only making proofs more verbose because you would typically need to include something along the lines of, "...since $N$ is an nhood of $a$ it contains an open set..." People who prefer generality at all cost may suggest the Singh definition is better.
 

Peter

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

Strictly speaking, Singh's definition is slightly more general. However, you can always convert a local basis in the sense of Croom to one in the sense of Singh and vice versa.

Croom to Singh

Since open sets about $a$ are nhoods of $a$, a local basis in the sense of Croom is automatically a local basis in the sense of Singh.

Singh to Croom

Let $\mathcal{B}^{S}_{a}$ be a local basis in the sense of Singh. Each element of $\mathcal{B}^{S}_{a}$ contains an open set about $a$. The collection of all such open sets is a local basis in the sense of Croom.


As you can see, whichever definition you choose is fine. Personally, I prefer the Croom definition because open sets are really what topology "boils down to" in my opinion. I cannot recall a time where working with nhoods provided a distinct advantage to working with open sets. Though it's been awhile since I studied general topology, I recall nhood definitions only making proofs more verbose because you would typically need to include something along the lines of, "...since $N$ is an nhood of $a$ it contains an open set..." People who prefer generality at all cost may suggest the Singh definition is better.

Thanks GJA ...

Really appreciate your help...

Peter