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The Jungle River metric induced topology compactness

Help sketching 'Jungle River Metric' in $mathbb{R}^2$Metric topology induced by the sum of two metricsFind sets of points, where function from one topological space to another is continuous.Proof with set compactness with river metrichighway metric topologically equivalent to euclidean metric?Show that the jungle river (barbed wire) metric is a metric3 homeomorphisms between spaces (2 with jungle metric)Prove that all three metrics induces the same topology on $X_1times X_2$Do these two metrics give rise to the same topology on $mathbb{R}^2$Are d1 and lift metric equivalent distances?

3

1

$begingroup$

I have the Jungle River metric induced topology, $tau$, given by the metric:

$$d(x,y) = begin{cases} |x_2-y_2|, & text{if $x_1 = y_1$;} \
|x_2| + |y_2| + |x_1-y_1|, & text{if $x_1 neq y_1 $} end{cases}$$

I need to prove that the topological space $(mathbb{R}^2,tau)$ is not compact.

Let $x=(x_1,x_2)$ then the balls are:

$$B_d={x_1}cdot(x_2-varepsilon,x_2+varepsilon) cup B_1((x_1,0), varepsilon-|x_2|)$$

where $B_1$ is the the ball with the $d_1$ metric.

I have problems finding a cover for $mathbb{R}^2$.

general-topology metric-spaces

share|cite|improve this question

edited May 9 '18 at 3:24

Narasimham

21k62158

asked May 9 '18 at 2:57

lore9696lore9696

464

$endgroup$

add a comment | 

3

1

$begingroup$

I have the Jungle River metric induced topology, $tau$, given by the metric:

$$d(x,y) = begin{cases} |x_2-y_2|, & text{if $x_1 = y_1$;} \
|x_2| + |y_2| + |x_1-y_1|, & text{if $x_1 neq y_1 $} end{cases}$$

I need to prove that the topological space $(mathbb{R}^2,tau)$ is not compact.

Let $x=(x_1,x_2)$ then the balls are:

$$B_d={x_1}cdot(x_2-varepsilon,x_2+varepsilon) cup B_1((x_1,0), varepsilon-|x_2|)$$

where $B_1$ is the the ball with the $d_1$ metric.

I have problems finding a cover for $mathbb{R}^2$.

general-topology metric-spaces

share|cite|improve this question

edited May 9 '18 at 3:24

Narasimham

21k62158

asked May 9 '18 at 2:57

lore9696lore9696

464

$endgroup$

add a comment | 

$begingroup$

I have the Jungle River metric induced topology, $tau$, given by the metric:

$$d(x,y) = begin{cases} |x_2-y_2|, & text{if $x_1 = y_1$;} \
|x_2| + |y_2| + |x_1-y_1|, & text{if $x_1 neq y_1 $} end{cases}$$

I need to prove that the topological space $(mathbb{R}^2,tau)$ is not compact.

Let $x=(x_1,x_2)$ then the balls are:

$$B_d={x_1}cdot(x_2-varepsilon,x_2+varepsilon) cup B_1((x_1,0), varepsilon-|x_2|)$$

where $B_1$ is the the ball with the $d_1$ metric.

I have problems finding a cover for $mathbb{R}^2$.

general-topology metric-spaces

share|cite|improve this question

edited May 9 '18 at 3:24

Narasimham

21k62158

asked May 9 '18 at 2:57

lore9696lore9696

464

$endgroup$

share|cite|improve this question

edited May 9 '18 at 3:24

Narasimham

21k62158

asked May 9 '18 at 2:57

lore9696lore9696

464

share|cite|improve this question

edited May 9 '18 at 3:24

Narasimham

21k62158

asked May 9 '18 at 2:57

lore9696lore9696

464

edited May 9 '18 at 3:24

Narasimham

21k62158

edited May 9 '18 at 3:24

Narasimham

21k62158

asked May 9 '18 at 2:57

lore9696lore9696

464

asked May 9 '18 at 2:57

lore9696lore9696

464

1 Answer
1

active

oldest

votes

1

$begingroup$

HINT:
Small open balls far away from the "river" are intervals.
There are uncountably many of such disjoint, open balls.
In particular, the plane with the "river" metric contains the uncountable discrete space, so it cannot be compact.

share|cite|improve this answer

edited Mar 12 at 21:18

Viktor Glombik

1,1821528

answered May 9 '18 at 3:02

Przemysław ScherwentkePrzemysław Scherwentke

11.9k52751

$endgroup$

add a comment | 

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1

$begingroup$

HINT:
Small open balls far away from the "river" are intervals.
There are uncountably many of such disjoint, open balls.
In particular, the plane with the "river" metric contains the uncountable discrete space, so it cannot be compact.

share|cite|improve this answer

edited Mar 12 at 21:18

Viktor Glombik

1,1821528

answered May 9 '18 at 3:02

Przemysław ScherwentkePrzemysław Scherwentke

11.9k52751

$endgroup$

add a comment | 

1

$begingroup$

HINT:
Small open balls far away from the "river" are intervals.
There are uncountably many of such disjoint, open balls.
In particular, the plane with the "river" metric contains the uncountable discrete space, so it cannot be compact.

share|cite|improve this answer

edited Mar 12 at 21:18

Viktor Glombik

1,1821528

answered May 9 '18 at 3:02

Przemysław ScherwentkePrzemysław Scherwentke

11.9k52751

$endgroup$

add a comment | 

$begingroup$

HINT:
Small open balls far away from the "river" are intervals.
There are uncountably many of such disjoint, open balls.
In particular, the plane with the "river" metric contains the uncountable discrete space, so it cannot be compact.

share|cite|improve this answer

edited Mar 12 at 21:18

Viktor Glombik

1,1821528

answered May 9 '18 at 3:02

Przemysław ScherwentkePrzemysław Scherwentke

11.9k52751

$endgroup$

share|cite|improve this answer

edited Mar 12 at 21:18

Viktor Glombik

1,1821528

answered May 9 '18 at 3:02

Przemysław ScherwentkePrzemysław Scherwentke

11.9k52751

edited Mar 12 at 21:18

Viktor Glombik

1,1821528

edited Mar 12 at 21:18

Viktor Glombik

1,1821528

answered May 9 '18 at 3:02

Przemysław ScherwentkePrzemysław Scherwentke

11.9k52751

answered May 9 '18 at 3:02

Przemysław ScherwentkePrzemysław Scherwentke

11.9k52751