CSIR-NET: Analysis/Topology Quiz 3


For π›Όβˆˆβ„ let βŒŠπ›ΌβŒ‹ denote the greatest integer smaller than or equal to 𝛼. Define 𝑑:ℝ×ℝ→[0,∞) by 𝑑(π‘₯,𝑦)=⌊|π‘₯βˆ’π‘¦|βŒ‹, π‘₯,π‘¦βˆˆβ„. Then which of the following are true?

d(x,y)=0, if and only if x=y, x,yβˆˆβ„
What can you say about d(0.2,0.4)?

d(x,y)=d(y,x), x,yβˆˆβ„
Clearly, since |x – y| = |y – x|, we can be sure that this option is true.

d(x,y) ≀ d(x,z) + d(z,y), x,y,zβˆˆβ„
What happens if x = 1.2, y = 2.2, z = 2?

d is not a metric on ℝ
Verifying option 3, triangular inequality is not satisfied

CSIR-NET December 2019, Part C Question.

Do you have any other ways to prove/disprove options above?

Beauty of the entire question is that you simply need to understand the definition of metric spaces well to find the answer. Watch the video below to learn more.


Recommended Books for Real Analysis

Click on the image to buy.

https://m.media-amazon.com/images/I/51KgfU+RvgL._SL160_.jpg
Introduction to Real Analysis by Bartle and Sherbert (4e)
https://m.media-amazon.com/images/I/31rJbDUVh7L._SL160_.jpg
Principles of Mathematical Analysis by Rudin
https://m.media-amazon.com/images/I/21xmoHk2GNL._SL160_.jpg
Real Analysis by Kumaresan
https://m.media-amazon.com/images/I/41icXR1neqL._SL160_.jpg
Introduction to Topology and Modern Analysis by Simmons
https://m.media-amazon.com/images/I/418FX7j0koL._SL160_.jpg
Topology of Metric Spaces

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