Metric Spaces Puns

I'm working on a HW problem where we've defined the distance between two subsets A, B of the same metric space (X, d) as the infimum over all possible d(a, b), where a ∈ A and b ∈ B. We are asked to prove that, given that A is compact and B is closed, if the two sets are disjoint, then d(A, B) > 0.

I've tried to prove it by contrapositive. Using a basic theorem about infimums, For every δ > 0, there is some a_δ ∈ A and b_δ ∈ B such that d(a, b) < δ. In the case where there are infinitely many a_δ, by the compactness of A, the a_δ have a limit point a_0 in A. I would like to show that a_0 is also a limit point of the b_δ, but the problem I'm running into is that even if I choose a very small radius for the neighborhood around a_0, the a_δ whose existence I'm given may be very far away from b_δ.

I am vaguely aware that there are solutions that use sequential compactness or continuity, but since neither have formally been introduced in the class, I'd prefer to avoid using those if at all possible.

My POV is of a physicist interested in the mathematical foundations of GR (and other metric theories of gravity) and whose knowledge of topology and real analysis is all self-taught and patchy. So I suppose "point set topology" is my only interest here, and that's what I'm asking about. I understand that pure mathematicians don't need a reason, they just like having algebraic structures to poke and prod, so if you're a pure mathematician just pretend you care about applications for a second.

I first was introduced to it as the study of "continuity of maps", and I learned how the topology 101 definition of a continuous map maps (heh) exactly to the epsilon-delta definition of continuity in real analysis. Then I went digging through real analysis and topology books and I think I've pieced together the following applied mathematician's/physicist's motivation of topology:

>With metric spaces we study the continuity of maps (and hence differentiability, which we need for physics) using a generic definition of "distance" (not necessarily Euclidean distance, or a vector inner product, or something like that). Topology is the study of continuity of maps at its most fundamental, i.e. without needing to invoke a concept of "distance", so that continuity can be studied in contexts more general that functions from R to R. In this way we can study differentiability with the absolute minimum of assumptions and extra structure.

Okay that all sounds fine, but it seems to me that any topological space we'd want to study would have a metric defined on it, and the open sets that make up a topology on a given set are usually chosen to be workable with a (generic) metric. And so it seems to me that the "minimal assumptions and minimal structure" thing is a bit misleading since we're really choosing open sets (or bases for them) with (generic) metrics in mind. So we've kind of built the metric space structure into our topologies with our choices of open sets. Or if not the full structure, the socket that it plugs into.

I'm worried that if (given a set) we constructed the open sets of a topology for it in some way not amenable to a metric space, then for that topology we'd have maps that would fit the definition of "continuous" but wouldn't be what anyone would actually call continuous, if presented with the map in isolation, but right now this is just a vague idea in my head that I haven't pinned down.

In which case, why not just talk about metric spaces all the

So I’ve been thinking about Cauchy sequences because we recently talked about it in analysis. I wonder if a Cauchy sequence in R (like just the usual metric |an-am|< epsilon) is also a Cauchy sequence in every other metric having a different perception of distance?

If we look on a function d: ℝ² -> ℝ with d(x,y) = √[(x-y)²], does that translate to √[(x1-y1)² + (x2-y2)² ] and if not, is it still a metric on ℝ ?

Last week, I finished my finals for my first Real Analysis course and was able to earn an A!

However if I am being honest, there is one thing I have not been able to overcome in this course: the Completeness property of metric spaces. I was fortunate this was not covered in the final exam.

I wish to spend my break reviewing and studying ahead for the next term.

>As far as the depth to which my course covered metric spaces, we:
>
>1) Defined what it means for a function to be a metric on a set.
>
>2) Defined what it means for a sequence in a metric space to be convergent and what it means for a sequence in the set to be a Cauchy sequence.
>
>3) Defined what it means for a metric space to be Complete.
>
>4) Proved the completeness of R^k and proved the Bolzano-Weierstrass Theorem for R^k .

My professor opted to skip the second half of this section which covered topology such as: interiors, open sets, closures, boundary points, open covers, and compactness. We were told that it was unnecessary for us as math undergraduates.

I provided all that information to give some idea as to my knowledge level of metric spaces. My question now is: what is completeness exactly?

Proving whether a metric space is complete is easily the hardest thing I've done in the course, I still am unable to interpret intuitively what we are showing, let alone constructing a proof. I understand the definition that Cauchy sequences in our set converges to an element in the set but what is the significance of doing so? Does this pave way for us to apply the Completeness Axiom in some generalized fashion? If so, what about the ordering of the metric space? Why do we choose the word "complete" to describe this property? Is there some sort of geometric interpretation that might help in visualizing completeness?

I recently created a website that shows, which cryptos are gaining most Twitter followers at the moment. So I’ve been trying to think when that information could be useful. I have a couple of ideas, and would love to hear if you have other ideas or arguments against (the Top 10 list at the bottom of this post).

1. Twitter growth is a unified metric that can be easily fetched for all projects. Metrics like blockchain wallet addresses, or monthly transactions or active users might not be readily available for all projects, and in some cases may not be comparable to other projects. Twitter growth could arguably be a proxy to the growth of the actual network usage (for sure not always the case though). Also, in one instance, I don't believe the proclaimed user numbers of a certain project, and looking at their twitter followers, it seems like the number they advertised may not be correct.
2. Twitter growth is not necessarily linked to market movements. Even when the market overall is bearish, a project may keep gaining more followers in the meantime, which might not be apparent from its market price movement.
3. If I had to guess the top 10 cryptos in terms of followers, I think my list would have looked quite different from reality. That indicates that there was something I didn't know about the crypto space.

However, one big problem of tracking the followers is fake accounts. Also, I think that projects with more following are those that are “easy to get” or that become a popular things like meme coins or games, whereas more technical projects don’t get as big a following. Therefore, the follower growth can look very different for the two types of projects.

I don’t believe that social media growth can be used alone to understand the crypto space. However, I feel like it might be a good tool in the toolkit to follow the trends in the crypto space. What do you think?

Here's the Top 10 list:

If you got that right, here's a bonus question: project which gained most new followers last week? Answer: &gt

Hi everyone,

I'm stuck on this metric spaces question. Any help is greatly appreciated:

Let X be a non-empty set and let d0 be the discrete metric on X. Under what circumstances is (X, d0) connected?

Thought process so far:

I have that it's only connected if X has 1 element and I can see why that is (AUB would not have A and B as disjoint, non-empty sets), but I don't know how to disprove a larger set X.

Thanks in advance!

Hi everyone, having a bit of trouble with this proof at the moment.

I need to change it to prove that a space with supremum norm and bounded real continuous functions is complete.

I have no idea how so any advice is really appreciated- thank you! :)

Hi everyone, here is a confusing question I am trying to solve on the subject of metric spaces:

Q: Let d_1 be the metric on the set C[a,b] of continuous functions from [a,b] to R, defined by:

d_1(f,g) = integral from a to b of |f(t)-g(t)|dt.

Prove that d_1 is a metric. Then, show that the same formula does not define a metric on the set of real-valued Riemann-integrable functions on [a,b]. (What the heck??)

Thank you in advance for any help! :)

I'm reading about metric learning in machine learning (e.g. https://arxiv.org/pdf/1812.05944.pdf) and I have a question I can't seem to find an answer to.

Suppose there is some dataset $\{x_{n}\}_{n=1}^{n=N}$ and someone gives you a pairwise distance matrix $D \in \mathbb{R}^{N \times N}$, where $D_{ij} = d(x_i, x_j)$ for some unknown metric $d(\cdot, \cdot)$. I would like to learn a function $f: \mathcal{X} \rightarrow \mathbb{R}^{F}$ (where $F$ is a predefined integer number of features) such that the following quantity is minimized:

$$\arg \min_{f}\sum_{a, b} \Big(D_{ab} - f(x_a)^T \big(\sum_{c,d}f(x_c) f(x_d)^T \big)^{-1} f(x_b) \Big)^2$$

In English, I'm asking how to learn a function $f$ such that if you compute its outputs on a given dataset, the function's pairwise distances (in the metric defined by the inverted empiric second moment matrix) match the given pairwise distances.

Has this been studied before? Does it have a known solution, or does anyone see a way to attack it?

I posted my question on Math Stackexchange, but I'm also posting here in case some people might know. Link is: https://math.stackexchange.com/questions/4308118/distance-metric-learning-optimizing-over-function-space

What do you guys think?

View Poll

Hi everyone, I have a short question here regarding metric spaces. I've no idea where to begin with it so any help is very much appreciated :)!

Q: Show that for any a,b,c,d in R, max(a+b, c+d) <= max(a,c) + max(b,d).

Looking to improve my topology.

Proposition Proof

I'm having an issue proving the statement in the title. In particular looking at the proof at https://ncatlab.org/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded I don't get how we can state that the sub-sequence doesn't converge given that the distance between two points of the sequence must be greater than epslion.

Thank you in advance

So fair warning - I don’t know of any elementary proof of the following fact (and neither does MO apparently...), despite its apparent simplicity. The existing proof methods are all pretty high powered. But the question statement itself is accessible if you know what a metric space is, so I was wondering if anyone wanted to give this a stab.

Problem set up:

Let X be a compact metric space. We say a continuous map T: X -> X is distal if inf (n) d(T^(n)(x), T^(n)(y)) = 0 implies that x = y.

From the definition one can see relatively easily that distal maps are always injective. Now the hard part:

Problem: Show that T is also surjective.

https://preview.redd.it/ykurz1081se71.jpg?width=1399&format=pjpg&auto=webp&s=c30f5a454a094b24a145381e16028d69c5e8f66e

An easily discernible fact of the cantor set $C = 2^N$ is that C is homeomorphic to two disjoint copies of itself, since $2^N = 2 x 2^N$. More generally, we have $n^N = n x n^N$.

Is there a converse to this? Suppose now that you have some arbitrary compact metric space X such that X is homeomorphic to n copies of itself. How "related" to $n^N$ is this space? For instance, X need not be homeomorphic to $n^N.$ $n^N x Y$, where $Y$ is any compact metric space, also has this self-similarity property as well... But are all such X of this form? that is, can they always be expressed as the product of some compact metric $Y$ and and $n^N$?

I need help with this proof.

"Let M be a metric space and let A be a subset of M and let C be a compact set. Prove that if the intersection of A with any compact set C is closed, then A is closed."

Let A ∩ C be a closed set.

How do I show that A must be closed? I know that since C is compact then every infinite set has a limit point, that is every sequence has a convergent subsequence.

Can someone help me?