2

Which of the following statements are true?

(a). Let $X$ be a set equipped with two topologies $\tau_1$ and $\tau_2$. Assume that any given sequence in $X$ converges with respect to the topology $\tau_1$ if, and only if, it also converges with respect to the topology $\tau_2$. Then $\tau_1 = \tau_2$.

(b). Let $(X, \tau_1)$ and $(Y, \tau_2)$ be two topological spaces and let $f : X \to Y$ be a given map. Then $f$ is continuous if, and only if, given any sequence $\{x_n\}$ such that $x_n \to x \in X$, we have $f(x_n) \to f(x) \in Y$.

(c). Let $(X, \tau )$ be a compact topological space and let $\{x_n\}$ be a sequence in $X$. Then, it has a convergent subsequence.

My attempsts ; all option a) ,b) and c) are all correct... by theorem of Arzelà–Ascoli theorem ...

Is my answer is correct or not ? and i would be more thankful who rectifying my mistakes........

José Carlos Santos
  • 440,053

1 Answers1

1

In $b)$ the $" \Leftarrow"$ part is not always true unless $(X,\tau)$ is a first countable space.

Here is a counterexample:

sequentially continuous on a non first-countable

$c)$ is not always true.Here is a reference for counterexample. The space $[0,1]^{[0,1]}$ which is compact from Tychonov's theorem.

https://www.andrew.cmu.edu/user/calmost/pdfs/sasms_F04.pdf

Marios Gretsas
  • 22,350