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I'm a beginner for a subject "topology".

While studying it, I found a confusing concept.

It makes me crazy..

I try to explain about it to you.

For a set X, I've learned that a metric space is defined as a pair (X,d) where d is a distance function.

I've also learned that for a set X, topological space can be defined as a pair (X,T) where T indicates a topology of X.

So, I thought that a metric space is not a topological space since second coordinate of a pair (X,d) is a distance function, so that it's different from the second coordinate of the pair (X,T) where T is a topology of X.

But, here is my question, in my book, it seems that author of the book treats a metric space as a kind of topological space. For example, he defined the definition of continuity of a function from a topological space to another topological space. But, later, he uses the concept of continuity to a function from a topological space to a metric space, too.

But, since I thought that a metric space is not a topological space, it makes me confused.

So, I want to ask you if a metric space is a topological space or not.