1
$\begingroup$

Consider the discrete topology $\tau$ on $X:= \{ a,b,c, d,e \}$. Find subbasis for $\tau$ which does not contain any singleton sets.

The definition of subbasis is as follows:

Definition: A subbasis $S$ for a topology on $X$ is a collection of subsets of $X$ whose union is $X$.

So let $S$ be equal to the collection of $\{a,b\}$, $\{c,d\}$ and $\{d,e\}$.

Clearly union of these three elements is $X$.

So should be $S$ - as defined - be taken as subbasis? Please check the answer I posted in comment.

$\endgroup$
  • 1
    $\begingroup$ $S$ will generate some topology on $X$, but will it generate the desired topology $T$ on $X$? $\endgroup$ – MPW Feb 22 '17 at 14:40
  • 1
    $\begingroup$ What about the singleton $\{b\}$? Your $S$ doesn't generate it. $\endgroup$ – ziggurism Feb 22 '17 at 14:40
  • 1
    $\begingroup$ You need that your sub-basis generates, by intersection, all the singletons of $X$ $\endgroup$ – Masacroso Feb 22 '17 at 14:45
  • $\begingroup$ Can I take S= collection of {a,b} , {b,c} , {c,d} , {d,e} , {a,e} as sub basis? $\endgroup$ – Kavita Feb 22 '17 at 16:07
  • 1
    $\begingroup$ One problem that may hinder your progress is that the "definition" you give of subbasis is not complete. $S$ is a subbasis of $\tau$ if $\{B \mid B \text{ is the intersection of finitely many elements of } S\}$ is a basis for $\tau$. It follows that the union of all the elements of a subbasis is $X$, but the latter alone is not enough. Clearly all the singletons give you a basis for $\tau$. Hence... $\endgroup$ – Fabio Somenzi Feb 22 '17 at 17:01
4
$\begingroup$

Hint: You can write $\{a\}$ as $\{a,b\}\cap\{a,c\}$. Do the same with each of the elements of $X$.

$\endgroup$
  • $\begingroup$ Can I take S= collection of {a,b} , {b,c} , {c,d} , {d,e} , {a,e} as sub basis? $\endgroup$ – Kavita Feb 22 '17 at 16:51
  • 1
    $\begingroup$ @Kavita: Looks good because you can get each point as an open set, so everything is open and the topology generated is indeed discrete. $\endgroup$ – MPW Feb 22 '17 at 17:54
0
$\begingroup$

The collection consisting of all $27$ non-singleton sets is a subbase: each of them is an open set (since the topology is discrete), and every singleton set can be expressed as the intersection of two non-singleton sets.

More economically, the five $4$-element sets constitute a subbase for the discrete topology; each singleton set is the intersection of the four $4$-element sets containing it.

Or just take the four sets $\{1,2,3\}$, $\{1,4,5\}$, $\{2,4\}$, $\{3,5\}$. Note that
$\{1\}=\{1,2,3\}\cap\{1,4,5\}$,
$\{2\}=\{1,2,3\}\cap\{2,4\}$,
$\{3\}=\{1,2,3\}\cap\{3,5\}$,
$\{4\}=\{1,4,5\}\cap\{2,4\}$,
$\{5\}=\{1,4,5\}\cap\{3,5\}$.

$\endgroup$
-1
$\begingroup$

The correct singleton sets should be $\{(a,b),(b,c),(c,d),(d,e),(a,e)\}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.