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Lebl-Basic-Analysis-I
· Problem 7.1.6
Metric Spaces
Mathematics · metric-spaces-definition
Problem
Let
(
X
,
d
X
)
(X, d_X)
(
X
,
d
X
)
and
(
Y
,
d
Y
)
(Y, d_Y)
(
Y
,
d
Y
)
be metric spaces. a) Show that
(
X
×
Y
,
d
)
(X \times Y, d)
(
X
×
Y
,
d
)
with
d
(
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
)
≔
d
X
(
x
1
,
x
2
)
+
d
Y
(
y
1
,
y
2
)
d\bigl((x_1, y_1), (x_2, y_2)\bigr) \colonequals d_X(x_1, x_2) + d_Y(y_1, y_2)
d
(
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
)
:
=
d
X
(
x
1
,
x
2
)
+
d
Y
(
y
1
,
y
2
)
is a metric space. b) Show that
(
X
×
Y
,
d
)
(X \times Y, d)
(
X
×
Y
,
d
)
with
d
(
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
)
≔
max
{
d
X
(
x
1
,
x
2
)
,
d
Y
(
y
1
,
y
2
)
}
d\bigl((x_1, y_1), (x_2, y_2)\bigr) \colonequals \max\{d_X(x_1, x_2), d_Y(y_1, y_2)\}
d
(
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
)
:
=
max
{
d
X
(
x
1
,
x
2
)
,
d
Y
(
y
1
,
y
2
)}
is a metric space.
Related theory
Continuous Maps Between Metric Spaces
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