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Normal books like Munkres tells us how the purely topological properties of metric spaces are preserved under homeomorphisms. But, I wish to know, is there any book which studies invariant properties of metric space which depend on both the topological and metric structure at the same time?

So, for instance, I would wish to know if a property like completeness is preserved under topologically equivalent metric spaces.

Edit: I am not sure why I am getting downvoted or why people think this is covered in a general Analysis text. I have checked Tao analysis-1 and 2, checked Kreyszig and currently reading Munkre's but I have not found in any of these book the topic I wish to check (at least a direct presentation).

If I am to be even more elaborate, it would be like so: I am looking for theorems on topological properties which translate into special metric properties but doesn't go backwards.

So, for example a properties like boundedness or Cauchy sequences.

Clemens Bartholdy
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2 Answers2

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One can think about the whole analysis as studying of the Euclidean metric and its variations. Mostly. This also opens doors to geometry and differential analysis. So any book on analysis will be good.

Another important object to study in this context is isometries: these depend on the metric, not on topology only.

Another context where metrical properties are studied a lot is data analysis.

So, for instance, I would wish to know if a property like completeness is preserved under topologically equivalent metric spaces.

It is not. $\mathbb{R}$ and $(0,1)$, with the Euclidean metric, are homeomorphic. But $(0,1)$ is not complete with respect to the Euclidean metric.

freakish
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  • I have tried analysis book but I felt they weren't focused directly to my goal. They seemed to be studying metric space in general rather than seeing these type of properties of metric space which depend on BOTH topology and metric structure – Clemens Bartholdy Sep 04 '22 at 15:46
  • @TrystwithFreedom so first of all, if you study metric properties than you implicitly study topological properties. It is not possible to study metric without topology, since metric induces a topology. Typically, when we study metric properties we often do it in the context of Euclidean metric, e.g. data analysis. – freakish Sep 04 '22 at 15:51
  • I understand what you mean but my question is slightly differnet . I have tried hard to elaborate more. – Clemens Bartholdy Sep 04 '22 at 17:22
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Perhaps the study of metrizable spaces is what you are looking for? A topological space $X$ is metrizable if there exists a metric $d$ on the set $X$ such that the given topology on $X$ is equal to the metric topology generated by $d$.

One way metrizable spaces are studied is by taking a property $P$ of metric spaces, and turning it into a topological property by making use of the existential quantifier. For example, despite the obvious fact pointed out in the comment of @MarkSaving that completeness is not a topological invariant, one can nonetheless define a Polish space to be a topological space $X$ which is homeomorphic to a complete metric space. So, for example, the metric space $(0,1)$ is not complete, but it is Polish, because it is homeomorphic to $\mathbb R$. My other favorite example of a Polish space (maybe everyone's favorite?) is the irrational numbers $\mathbb I = \mathbb R - \mathbb Q$. The study of Polish spaces is an important subtopic in the general study of metrizable spaces.

Lee Mosher
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  • Let me try be more specific. I am looking for properties at the intersection of the metric properties of a metric space and topological properties of a topological space. So, for example a property like boundedness or cauchy sequences. – Clemens Bartholdy Sep 04 '22 at 17:10
  • Well, boundedness of a subset is a purely metric property which has no topological analogue, for the following simple reason: every metrizable topological space has a bounded metric. – Lee Mosher Sep 04 '22 at 17:15
  • Also, as you know from the responses to your own recent posts, the property that a sequence be a Cauchy sequence is not a topological property; nonetheless there are additional structures added to a topological space which are used to abstract the Cauchy property. – Lee Mosher Sep 04 '22 at 17:17
  • Perhaps your quest could be satisfied by delving more deeply into the topics that have come up in this post and previous posts. – Lee Mosher Sep 04 '22 at 17:19
  • Ah okay, I see, I guess the issue was my formulation. I mean, compactness of metric space is actually topological. But the same time, this compactness reduces to something which only works in the metric space (boundedness). – Clemens Bartholdy Sep 04 '22 at 17:19
  • And on the previously mentioned things, I am trying to make a road map on things I can possibly explore later on. I have been reading the theory from Munkre's on and off and trying to think about it. Most of the questions are just checks if the intuition I am getting from speculating is correct or not. I'm a bit tight on time these days due to circumstance but I hope to pursue when I get more time. Thanks for the engagement and help btw. – Clemens Bartholdy Sep 04 '22 at 17:29