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I am trying to prove that the $\sqrt[3]{x}$ is continuous using topology.

The question is:

$f : (0, \infty) \rightarrow (0, \infty)$

$x \rightarrow \sqrt[3]{x}$

Use the definition of a continuous function in terms of pre-images of open subsets to prove that $f$ is continuous in $(0, \infty)$.

I know that the function $F : X \rightarrow Y $ is continuous if the inverse image of every open subset of $Y$ is open in $X$.

So I assume I need to prove that any $[(a,b)] \in Y $ is an open set?

If so, how would I begin to prove this?

pc799
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  • Which are the domain and the codomain of $F$? – Another User May 14 '22 at 16:43
  • @AnotherUser - The domain is $[0, \infty]$ and the range I assume is the same although it wasn't explicitly stated in the question to me. – pc799 May 14 '22 at 16:53
  • I made no reference to the range; I wrote about the codomain instead. And are you sure that $\infty$ belongs to the domain? – Another User May 14 '22 at 17:17
  • Sorry, I meant the codomain. Although it was never stated in the question. And yes, the domain definitely includes $\infty$ – pc799 May 14 '22 at 17:21
  • Since you don't know what is your codomain, then you cannot possibly know its topology. Also, you did not tell us which topology you are taking into account in $[0,\infty]$. Therefore, there is not enough information to solve the problem. – Another User May 14 '22 at 17:36
  • The domain is $[0, \infty]$ and the codomain is $[0, \infty]$. And the questions asks me to prove it in terms of pre-images of open subsets of the range to prove it is continuous in $[0, \infty]$. – pc799 May 14 '22 at 19:42
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    I believe you're being a bit misguided. You need to first say what you choose as open sets in the domain and what you choose as open sets in the domain. The choice of topology also induces the choice of continous function between the two spaces. – tryst with freedom May 14 '22 at 19:49
  • https://math.stackexchange.com/questions/687414/prove-continuity-for-cubic-root-using-epsilon-delta – tryst with freedom May 14 '22 at 19:57
  • @Aplateofmomos - I have done the proof using epsilon-delta. The second part of the question (edited the post to show that) asks me to prove the same thing (continuity) but using the definition of a continuous function in terms of pre-images of open subsets of the range. I am very new to this and so I don't quite understand what I should be doing for this question. The way I understood it was if I could prove that any subset of the range was an open subset then any subset of the domain would be open too by definition? – pc799 May 14 '22 at 20:01
  • I feel you didn't understand my point. You have to say what the open sets are. When you do that, you also are saying what the topology is. Different topology makes different maps continous. – tryst with freedom May 14 '22 at 20:02
  • @Aplateofmomos - How should I go about choosing an open set? – pc799 May 14 '22 at 20:04
  • https://math.stackexchange.com/questions/2234571/examples-of-topologies-on-r – tryst with freedom May 14 '22 at 20:05

1 Answers1

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Let $(a,b)\subseteq(0,\infty)$.Then $f^{-1}((a,b))=(a^3,b^3)$, which is an open set in $(0,\infty)$. Since $\mathcal{B}=\{(a,b):a,b\in (0,\infty)\}$ is a basis for the subspace topology on $(0,\infty)$ induced by the usual topology on $\mathbb{R}$, this suffices to show that $f$ is continuous. (We showed that the preimage of every basis element for the topology of the codomain is open in the domain of $f$)

closedrhombus
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