Topology induced by a metric?

  1. When we say that a metric space (X,d) induces a topology or "every metric space is a topological space in a natural manner" we mean that:
    A metric space (X,d) can be seen as a topological space (X,τ) where the topology τ consists of all the open sets in the metric space?
    Which means that all possible open sets (or open balls) in a metric space (X,d) will form the topology τ of the induced topological space?
    Is that correct?
  2. jcsd
  3. quasar987

    quasar987 4,773
    Science Advisor
    Homework Helper
    Gold Member

    Well, yes, but what are the open sets in (X,d)?

    They are those sets A such that for every a in A, there is a r>0 such that B(a;r)={x in X : d(x,a)<r} is entirely contained in A.

    You can verify that these sets form a topology on X.
  4. Saying: "or open balls" is incorrect, the rest is correct. We say that a topology T on a space X is induced by a metric d on X iff the open balls generated by d forms a BASIS for the topology T (i.e. a set U is open iff it's a union of open balls).
  5. Oh I see, that's because the collection of open balls is a subset of the collection of all open sets.
    It makes sense, thank you both for your time!
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