# What is a metric space?

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A metric space is a topological space in which we can define a consistent notion of distance. If you don't know what "topological space" means, don't worry about it--just think of it as a set of points.

Distance in this case is very abstract and it may be very different from the ordinary ("Euclidean") notion of distance. However, it's not *that* abstract--we require that the distance have certain properties in common with our ordinary conception of distance so that things don't get too meaningless. These properties are the following:

1) The distance between two points is never negative--that just wouldn't make sense!

2) The only pairs of points (x,y) such that the distance between x and y is zero are those where x and y are actually the same point. So if x and y are different points, there is some positive distance separating them.

3) Calculating the distance between points doesn't matter in which order you take the points.

4) The last property is the so-called triangle inequality. It's derived, as the name may suggest, from the properties of triangles. Let A, B, and C be the three sides of a right triangle in the metric space, and suppose C is the hypotenuse. The triangle inequality is just that the hypotenuse C can't be longer than the lengths of A and B added together.

I hope this helps! :D

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• Simple answer: it's a bunch of points where we can tell the distance between each pair of points.

For an explanation of what I mean by "distance", we'd have to get to the "more complicated answer" - heh.

MInor correction for alwbsok:

"d" would be a metric on M, not on M x M... it's a function on M x M, but it's not a metric on M x M because we're not talking about finding the distance between two points in M x M.

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• I had no idea, until I searched it on Wikipedia:

http://en.wikipedia.org/wiki/Metric_space

I'm first year Uni student, so some of the specifics went a little over my head, but I got this:

"d" is a metric on the set M x M if, for any x,y, and z in M, the following properties are satisfied:

d(x,y) >= 0

d(x,y) = 0 <==> x = y

d(x,y) = d(y,x)

d(x,z) <= d(x,y) + d(y,z)

M is a metric space if there exists a metric (a "d") on it.

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• It is a simple and rather nice type of topological space in which it makes sense to talk about the distance between any pair of points. This "distance" may not be the usual distance at all, but it must have certain distance-like properties, such as, the distance between any point and itself is always 0, the distance between two points is never negative, the triangle inequality is satisfied. This is not a precise definition but should be enough to give you the idea.

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• Any discrete area is metrizable via using the metric d(x,y) = a million - delta(x,y), the place delta(x,y) is the Kronecker delta. So, evaluate the genuine numbers with the discrete topology as a metric area, and since the uncountable set of things. In a discrete area, no element is a decrease element of any set, for the reason that {x} is an area of x, and contains no factors different than x mendacity in any set. (specific, that made experience in case you unwind the definition of decrease element and the certainty i'm universally quantifying over all instruments.)

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