Metric space versus Topological space

[infinityQ]

1. Any metric space can be converted into a topological space such that an open ball in a metric space corresponds to a basis in the corresponding topology (metric spaces as a specialization of topological spaces ).
2. Any topological space can be converted into a metric space only if there is a metric d such that the topology induced by d corresponds to an original topology.

I am wondering if above statements are true or not.

As per #2, if #2 is right, what topological properties with which every #2 convertible (metrizable ?) topological space share?

Thanks in advance.

 

[morphism]

This is really just a matter of looking at the definitions. What does it mean for something to be a metric space? A topological space? What is the topology "induced" by a metric?

 

[Crosson]

This is really just a matter of looking at the definitions. What does it mean for something to be a metric space? A topological space? What is the topology "induced" by a metric?

The statement of necessary and sufficient conditions for an arbitrary topological space to be metrizable go way beyond looking at the definitions, see http://en.wikipedia.org/wiki/Metrization_theorems" for a starting point.

 

[morphism]

Where does the OP ask for necessary and sufficient conditions for metrizability?

Is it in the following passage:

As per #2, if #2 is right, what topological properties with which every #2 convertible (metrizable ?) topological space share?

Because I couldn't comprehend what he/she was asking here.

 

[infinityQ]

What I was curious about is that if we can convert two kind of spaces (both metric and topological spaces) each other in a systematic way, we may reduce one space's problem into another space's problem.

My previous assertions were as follows:
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1. Any metric space can be converted into a topological space such that an open ball in a metric space corresponds to a basis in the corresponding topology (metric spaces as a specialization of topological spaces ).
2. Any topological space can be converted into a metric space only if there is a metric d such that the topology induced by d corresponds to an original topology.
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After reading some wiki, my tentative conclusion is
For #1 (metric space to topological space), I think it is a true statement by definition, so we can convert any metric space into a topological space.
For #2 (topological space to metric space), I found it is possible for only limited cases and those convertible (metrizable ?) topological spaces share some topological properties like Hausdorff and first countable.

please let me know if something is wrong for above two assertions.

 

[HallsofIvy]

Yes, a "metric space" is a specific kind of "topological space". It is not a matter of "converting" a metric space to a topological space: any metric space is a topological space. There exist topological spaces that are not metric spaces. Example: any set, X, with the "indiscreet" topology: X itself and the empty set are the only "open" subsets of X.