Question

Let X be any infinite set and τ be the discrete topology on X. Find a subbasis S for τ such that S does not contain any singleton sets.

Answer 1

A subbasis for the discrete topology on an infinite set X that contains no singleton sets can be formed by the collection of all two-element subsets of X, as their intersections can generate the entire topology.

Step-by-step explanation:

When considering an infinite set X with a discrete topology τ, a subbasis S for τ is a collection of sets whose finite intersections generate the topology. Since τ includes all subsets as open sets, any collection of sets that covers X can be used to form the subbasis. However, creating a subbasis without singleton sets requires us to choose sets that are not singletons but still cover the entire set X.

To construct such a subbasis, we can consider the collection of all two-element subsets of X. This collection clearly does not contain any singleton sets. The intersections of these two-element sets will include singletons as well as sets with more elements, which can generate the entire discrete topology on X. Thus, the collection of all two-element subsets of X is a valid subbasis for the discrete topology on X that contains no singleton sets.

Answer 2

A subbasis for the discrete topology on an infinite set X that does not include any singleton sets can be constructed by taking the complements of all possible finite subsets of X. Since X is infinite, the complements will not be singletons and their finite intersections can generate all subsets, thereby forming the entire discrete topology.

Step-by-step explanation:

You are asking about constructing a subbasis for a topology, specifically for the discrete topology on an infinite set, where the subbasis does not include any singleton sets. In topology, a subbasis S for a topology τ is a collection of sets whose union equals the entire space X, and the finite intersections of which form a basis for the topology τ.

To devise such a subbasis for the discrete topology on an infinite set X, we can consider the collection of all possible finite complements in X. Since X is infinite, the complement of any finite subset is itself an infinite set, which would satisfy the condition of avoiding singleton sets.

Here's how one might construct such a subbasis:

1. Consider all finite subsets F of X.
2. For each finite subset F, take the complement in X, denoted as X \ F.
3. The collection S of all such complements will serve as the subbasis.

This collection S of complements of finite subsets will indeed form a subbasis, because the discrete topology on X is generated by all subsets, and finite intersections of complements of finite subsets can produce any subset, including singletons, thereby generating the entire discrete topology.