1. ## Questions

Q: removed

2. Originally Posted by johndon
Q1: Let ℝ be given the usual topology. Find a function f: ℝ→ ℝ
such that f is continuous at x = 0 , and f is discontinuous at all x ≠ 0.
The "Dirichlet function" is a famous example. Let f(x)= 0 if x is irrational and f(x)= 1/n where x= m/n reduced to lowest terms.

Q2: Define a metric d on ℝ so that the sequence converges to
1.
What sequence?

Q3: Define a topology τ on ℝ so that the sequence Xn=1/n converges
to 0 , and also converges to 1, but does not converge to any other number. Is
τ metrizable?
Take T to consist of all open sets of R (under the usual topology) except that "1" is added to any set containing "0" and "0" is added to any set containing "1".

3. ## missing sequence

the sequence is Xn=1/n in Q2

4. Here is for the second part of question 3

Is this space metrizable?
Hint:
Spoiler:

Theorem: Let $\displaystyle X$ be Hausdorff, then if $\displaystyle \{x_n\}_{n\in\mathbb{N}}$ is a convergent sequence in $\displaystyle X$, then it's limit $\displaystyle x$ is unique.

Proof: Suppose that $\displaystyle x\ne y$ and let $\displaystyle U_x,U_y$ be the respective disjoint neighborhoods of $\displaystyle x,y$ guaranteed by Hausdorffness. Now, since $\displaystyle x_n\to x$ all but finitely elements of $\displaystyle K=\{x_n:n\in\mathbb{N}\}$ lie in $\displaystyle U_x$ and so not all but finitely man values of $\displaystyle K$ may lie in $\displaystyle U_y$. It follows that $\displaystyle x_n\not\to y$. The conclusion follows.

Soo..................

Originally Posted by johndon
the sequence is Xn=1/n in Q2
Work? Where is it?