# Solving Problems in French Railway Metric

• cummings12332
In summary, The user is asking for help with solving problems related to open sets in a topology induced by the French railway metric in R2. They specifically want to know if A=(0,1)× R and B=(-1,1)× R are open in this topology. They suggest using the fact that A is a union of metric balls and describe the open balls for A. They then ask if this can be generalized to other centers and radii. Finally, they ask if choosing the radius of the open ball to be r=min{1-x1,x1} would prove that A is open.
cummings12332

## Homework Statement

can someone help me to solve these problems in details??
Consider A =(0,1)× R. Is A open w.r.t. the topology induced by the French railway metric in R2? how

2. The attempt at a solution
I know A is open in the topology induced by d if and only
if U is a union of metric balls. But for my questions here, how can I see that A is a union of metric balls?? should I take any x in A then exists r>0 s.t. B(x,r) is open in A?but now the French railway metric gives me 2 different cases, how to consider?

An obvious first thing to do is to describe the open balls. What do the open balls look like?

micromass said:
An obvious first thing to do is to describe the open balls. What do the open balls look like?

If I choose the centre be (1,1) and radius 1 to be the open ball then it is segment with length 2 , 45degree to x-axis and the midpoint of the segment is (1,1)

cummings12332 said:
If I choose the centre be (1,1) and radius 1 to be the open ball then it is segment with length 2 , 45degree to x-axis and the midpoint of the segment is (1,1)

Go on. Can you generalize this to other centers and radii?

micromass said:
Go on. Can you generalize this to other centers and radii?

Can I choose the radius of open ball be r=min{1-x1,x1} then for all x=(x1,x2) in A we have B(x,r) is contained in A so A is open??

Last edited:

## 1. What is the French Railway Metric?

The French Railway Metric, also known as the French Railway Gauge, is a standard gauge used in France and other European countries for railway tracks. It has a track gauge of 1,435 mm, which is slightly wider than the standard gauge used in the United States and other countries.

## 2. What are some common problems encountered in the French Railway Metric system?

One of the most common problems encountered in the French Railway Metric system is the compatibility issue with other railway systems that use a different gauge. This can cause delays and disruptions in international train travel. Another problem is the limited availability of trains and equipment designed specifically for the French Railway Metric, making it challenging to expand the railway system.

## 3. How do scientists solve problems in the French Railway Metric?

Scientists use various methods and techniques to solve problems in the French Railway Metric, including mathematical modeling, computer simulations, and physical testing. They also collaborate with engineers and other experts to develop innovative solutions and improve the efficiency and reliability of the railway system.

## 4. What are some potential solutions to the compatibility issue in the French Railway Metric system?

One potential solution is the use of variable gauge trains, which can adjust their wheelsets to different track gauges. Another solution is the construction of dual gauge tracks, which allow trains of different gauges to share the same track. Additionally, there have been discussions about adopting a global standard gauge to improve compatibility among different railway systems.

## 5. How does the French Railway Metric contribute to sustainable transportation?

The French Railway Metric is considered a more sustainable mode of transportation compared to road and air travel. It produces fewer greenhouse gas emissions and consumes less energy. Additionally, the railway system can be electrified, further reducing its environmental impact. The French government has also invested in high-speed trains, which are more energy-efficient than conventional trains, making them a greener option for long-distance travel.

Replies
1
Views
1K
Replies
19
Views
2K
Replies
1
Views
1K
Replies
6
Views
1K
Replies
6
Views
4K
Replies
5
Views
2K
Replies
15
Views
2K
Replies
14
Views
2K
Replies
10
Views
2K
Replies
3
Views
2K