🛈 This materials is useful for Unit 1 in CSIR-NET.

**Definition:** , where

Prove that is a metric on , as an exercise.

We denote it as .

In terms of topological spaces, , where is the power set of

*Some Interesting ***Properties:**

*Some Interesting*

**Properties:**

- Every function is continuous in .
- Every set is open subset of .
- Every set is closed in .
- is not complete.

**Examples of interest**

**Examples of interest**

- greatest integer smaller than or equal to x is not continuous in with usual metric, but is continuous in .
- is not open in , but it is open in . In fact, every set is open in
- is closed in , and in .

##### Challenges

- Can you find an example of compact set in ?
- Can you characterize convergent sequences in ?

Leave your thoughts in the comments below!

## Recommended Books for Real Analysis

Click on the image to buy.