This is a homework problem in my algebra analysis class. I'm not really sure where to begin. I need to prove the statement below.
Y X where X is a metric space with function d. Then (Y,d) is a metric space with the same function d. How can I show that X and Y have the same metric function?
this definition of a metric, then you see that the function does not loose any of its properties when applied to points of .
strictly speaking you should speak of a separate function d*, the restriction of d to YxY. this is because (in some views) the domain and co-domain of a function are part of the function's definition. in other words, it's not just "the rule" that defines a function, it's also "what you apply the rule TO", and "where the values of the function LIVE".
but...it is common practice to make no distinction between the surjective function
and the "general function" it came from:
f:A→B. be aware of this...it can trip you up sometimes.
of course d* "is really" just d, because:
d*(y1,y2) = d(y1,y2), where on the right, we are considering the y's as elements of X.
in general, any statement of the form:
FOR ALL x,y,z in X: property (insert something here) is true, remains true even if we pick x,y and z from a SUBSET of X.
informally speaking, if two points lie within "an epsilon ball" in X, and both points are in Y, then they are in "an epsilon ball" in Y (but: "epsilon balls" in Y aren't always "round", parts of them might be "chopped off" because they lie in the part of X outside Y).