Prove that distance function on metric is continuous
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To show that d −1 (a,b) d−1(a,b)
is open, you need to specify for each point (x,y) (x,y)
in it a basic open set of the product topology (yes, we should use properties of the product topology somewhere), i.e. a set of the form U×V U×V
with u,V u,V
open, x∈U x∈U
, y∈V y∈V
. As X X
is a metric space, we may try open balls U=B r (x) U=Br(x)
, V=B r (y) V=Br(y)
for suitable r r
. How can we choose r>0 r>0
to enforce B r (x)×B r (y)⊆d −1 (a,b) Br(x)×Br(y)⊆d−1(a,b)
? (Simply translate what this means) You will need (alas!) the defining properties of metric for this.
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