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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.

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    $S$ will generate *some topology* on $X$, but will it generate the *desired topology* $T$ on $X$? – 2017-02-22
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    What about the singleton $\{b\}$? Your $S$ doesn't generate it. – 2017-02-22
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    You need that your sub-basis generates, by intersection, all the singletons of $X$ – 2017-02-22
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    Can I take S= collection of {a,b} , {b,c} , {c,d} , {d,e} , {a,e} as sub basis? – 2017-02-22
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    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... – 2017-02-22

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

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    Can I take S= collection of {a,b} , {b,c} , {c,d} , {d,e} , {a,e} as sub basis? – 2017-02-22
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    @Kavita: Looks good because you can get each point as an open set, so everything is open and the topology generated is indeed discrete. – 2017-02-22