Comparing Open Bases and Covers

Question

In Topology, I see a resemblance and similarity between open bases and open covers. Although this is a short question,

what is the defining difference between the two that sets them apart? Examples?

Answer 1

An open cover of a set is a collection of sets whose union contains the set. A basis is a collection of sets for which every open set is a union of elements of the basis.

Answer 2

A base must be able to produce any open set by taking unions of sets in the base; an open cover only needs to cover the whole space. It's easy to see that a base must be an open cover, but an open cover need not be a base.

For example, take the standard topology on the real numbers $\mathbf{R}$ and define the set $\mathcal{S} =\{\mathbf{R}\}$. $\mathcal{S}$ is an open cover because each set in $\mathcal{S}$ is open (there's only one set in $\mathcal{S}$) and the union of all the sets in $\mathcal{S}$ is the whole space $\mathbf{R}$. But $\mathcal{S}$ is not a base: for example, the open interval $(0, \infty)$ is an open set in this space, but you can't get that open set as a union of sets in $\mathcal{S}$.

EDITED TO ADD: Intuitively, a base must include very small open sets, because those sets must get down inside every neighborhood of every point. An open cover doesn't need to do that, as long as its sets are large enough to cover the whole space.

Answer 3

For an open base,

You must be able to generate every open set in the topological space(X,T) with the help of the union of some or all of it's sets.

But for an open cover

It would be enough if the union of all it's sets equal the set X.

Like it can be seen, there is a stricter condition placed on an open base

• Since X itself is an open set, every open base will be an open cover.

• And since not every open set of (X,T) might be exactly generated by the union of some sets of an open cover, an open cover is not always an open base. By exactly, I mean, the union of some sets in open cover must equal the open set, nothing more nothing less.