What is a topological space

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2.9.6 and as a topological space.

Let us summarize the properties of the open sets. Let it be a metric space and be the family of all open sets in. Then applies

Definition 2.9.27. A family of subsets of a basic set (i.e.) which fulfills the properties (2.50) - (2.52) is called a topology, the pair is called a topological space. The elements are subsets of what are called environments.

The family of open sets of a metric space always forms a topology and thus gives the structure of a topological space. The topology created in this way on a metric space is called the canonical or the induced topology. The elements of this topology, i.e. the surroundings, are nothing other than the open sets of metric space.

Example 2.9.28. An important example of topological spaces. Since and are metric spaces, a corresponding topological structure is induced in each case. The environments in these topologies are the open sets in and. A set is open in and, if each of its points also contains a nontrivial open sphere,,.

Problem 2.9.29. At least the so-called trivial topology can be introduced on any set. Show that the axiomatics of topology is fulfilled. However, this choice of topology often proves to be unsuitable for applications. A trivial distance function for and for can also be defined on each set. Show that the axiomatics of a distance function satisfies and that this trivial metric induces the trivial topology on!

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