It can be shown that given a basis, T C indeed is a valid topology on X.

In our previous example, one can show that Bsatis es the conditions of being a basis for IRd, and thus is a basis generating the topology Ton IRd. The basic idea is that a basis is the collection of all finite intersections of sub-basis elements. This is typically the case for the Zariski topology on the spectrum of a ring.
A subbasis for a topology on is a collection of subsets of such that equals their union. We refer to that T as the metric topology on (X;d). (2) For each x ∈ X and each basis element B ∈ B containing x, there is a basis

Finally, suppose that we have a topological space . Subspace topology. (i)One example of a topology on any set Xis the topology T = P(X) = the power set of X(all subsets of Xare in T , all subsets declared to be open).

(Standard Topology of R) Let R be the set of all real numbers. Bases have been introduced because some topologies have a base consisting of open sets that have specific useful properties. The topology generated by is finer than (or, respectively, the one generated by ) iff every open set of (or, respectively, basis element of ) can be represented as the union of some elements of .

The basis consisting of all the singletons in a set X(Example2.3.1) generates the discrete topology on X.

Basis for a Topology 4 Lemma 13.3. Example 1.1.9. So, the open sets in a topology are all possible unions of finite intersections of sub-basis elements. We can also get to this topology from a metric, where we define d(x 1;x 2) = ˆ 0 if x 1 = x 2 1 if x 1 6=x 2 T. See the 2 2. The basis consisting of all the open intervals in R (Example2.3.3) generates the usual topology on R. We can actually \do better" than this basis, in a certain sense. Let B and B0 be bases for topologies T and T 0, respectively, on X. 13. Example 1.7. In mathematics, a base B of a topology on a set X is a collection of subsets of X that is stable by finite intersection. I hope that answers your question! basis of the topology T. So there is always a basis for a given topology. a topology T on X. A base defines a topology on X that has, as open sets, all unions of elements of B. Let Bbe the collection of all open intervals: (a;b) := fx 2R ja
The open sets in a topology are all possible unions of basis elements.