# To find subbasis for a topology on X.

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.

• $S$ will generate some topology on $X$, but will it generate the desired topology $T$ on $X$? – MPW Feb 22 '17 at 14:40
• What about the singleton $\{b\}$? Your $S$ doesn't generate it. – ziggurism Feb 22 '17 at 14:40
• You need that your sub-basis generates, by intersection, all the singletons of $X$ – Masacroso Feb 22 '17 at 14:45
• Can I take S= collection of {a,b} , {b,c} , {c,d} , {d,e} , {a,e} as sub basis? – Kavita Feb 22 '17 at 16:07
• 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... – Fabio Somenzi Feb 22 '17 at 17:01

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

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

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
$$\{1\}=\{1,2,3\}\cap\{1,4,5\}$$,
$$\{2\}=\{1,2,3\}\cap\{2,4\}$$,
$$\{3\}=\{1,2,3\}\cap\{3,5\}$$,
$$\{4\}=\{1,4,5\}\cap\{2,4\}$$,
$$\{5\}=\{1,4,5\}\cap\{3,5\}$$.

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