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We have the following definition for a Cauchy sequence $\{p_n\}_{n\in\mathbb{N}}$ in $\mathbb{R}^n$: $$\forall\varepsilon>0~\exists N_\varepsilon\in\mathbb{N}~\backepsilon\forall m,n\ge N_\varepsilon,~\mathrm{d}(p_n,p_m)<\varepsilon$$ I came across something regarding this online that read,

Being a Cauchy sequence is not a topological notion: let $X=(0,1),~Y=(1,\infty),$ $f:X\rightarrow Y,p\mapsto\frac{1}{p}$ and $p_n=\frac{1}{n}$. Then $p_n$ is a Cauchy sequence, but $f(p_n) = n$ is not even bounded so cannot be a Cauchy sequence.

I do not understad what this paragraph means. What is meant by Cauchy sequence not being a topological notion?

reyna
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    Continuous mappings are the morphisms of between the topological spaces. A topological property should preserve under them. – fantasie Oct 22 '20 at 13:00
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    @fantasie Shouldn't they be homeomorphisms? – Maximilian Janisch Oct 22 '20 at 13:00
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    This description can be a bit unsatisfying, but I basically think of "topological property" as "thing that is preserved by application of a homeomorphism". In this case, $f$ is a homeomorphism, $p_n$ is Cauchy, and $f(p_n)$ is not, so Cauchiness isn't a topological notion. – Ian Oct 22 '20 at 13:01
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    Related https://math.stackexchange.com/questions/3839837/cauchy-sequence-is-not-a-topological-notion?rq=1 – red whisker Oct 22 '20 at 13:03
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    @MaximilianJanisch Preserving under homeomorphisms are of course more solid rules that you shouldn’t violate, but I suppose that to preserve under continuous maps means something. (Although I admit that homeomorphisms seems more suitable here.) My point is, continuous mapping is like everything that topology ever talks about. If Euclidean geometry is about properties that are invariant under the group isometries, and projective geometry to the projective transformations, then the corresponding thing for topology ought to be a certain type of continuous maps. – fantasie Oct 22 '20 at 13:13
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    @fantasie In general, one cannot expect a non-invertible map to preserve any property. You really want invertible maps to do that. Also note that your examples are all invertible. – asdq Oct 22 '20 at 14:09

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Topological notion is a notion that can be preserved by a homeomorphism, which is a continuous map and its inverse map is also continuous.

In your case, $\{p_n\}$ is a Cauchy sequence and $f$ is a homeomorphism, but $\{f(p_n)\}$ is not a Cauchy sequence, which implies Cauchy sequnce is not a topological notion.

Nate
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    preserved by homeomorphisms, right? – red whisker Oct 22 '20 at 13:02
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    Just continuity wouldn't necessarily preserve a topological property; for example, the continuous image of an open set isn't generally open, but openness is definitely a topological property. – Ian Oct 22 '20 at 13:02