Prove the Inverse of a Nonconstant Harmonic Function is Unbounded

Let u be a nonconstant harmonic function on C. Show that for any c∈R,u−1(c) is unbounded. Hint: {|z|>R} is connected for any R>0. It seems like this proof might require some sort of a “trick”, because I’m not sure how to attack this one directly. I know what I want to show is that for … Read more

Maximum and Minimum of Totally Ordered Compact Sets

Let T be a totally ordered compact set. Does this always imply that a maximum and a minimum element of T exist under this total order? And if not, what about the special case where: T is a closed and bounded interval of the real line (ergo compact) with the usual total order ≤ of … Read more

The inverse image of dense set is dense and of a comeager set is comeager?.

Let $X,Y$ be topological and $f:X \to Y$ be open and continuos. I am studying Baire space and I would like to try the following facts : $(i)$ The inverse image of dense set is dense. $(ii)$ The inverse image of comeager set is comeager. I am studying Baire space and I would like to … Read more

Embedding R\mathbb{R} into S2S^{2}

Does there exist an embedding f:R→S2 with a closed image? I believe not, but I’m stuck with how to prove that. It would be nice to hear several different proofs if my guess is true. Answer Let f:R→S1 topological embedding and let R=f(R). If R is closed in S1 R is compact and f is … Read more

Property of Hausdorff spaces

I want to show that $$\text{E is a Hausdorff space} \iff \bigcap_{V(x) \text{ is a closed neighborhood of X}} V(x)=\{x\}$$ I am sure my proof is incorrect, since along the way, I have managed to proove that $$\text{E is a Hausdorff space} \iff \bigcap_{V(x) \text{is a neighborhood of X}} V(x)=\{x\}$$ and it must be a … Read more

First uncountable ordinal

I am a beginner of ordinals and I don’t know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this. Let $X$ be a set of uncountable cardinality. Using the Principle of Well Order we have a well ordering $\le$ on X(and $<$ means $\le$ but not … Read more

Prove that exist bijection between inverse image of covering space [duplicate]

This question already has answers here: If h:Y→X is a covering map and Y is connected, then the cardinality of the fiber h−1(x) is independent of x∈X. (3 answers) Closed 2 years ago. Let B be path-connected and p:E→B covering map (with E as covering space). Prove that ∀a,b∈B exist 1-1 injection correspondence between p−1(a) … Read more

Regular and non-regular covering spaces of S1∨S1∨S1 \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} .

I tried to draw the regular and non-regular covering spaces of S1∨S1∨S1. I think the regular covering space is: Is it true? How do you draw the non-regular covering space of this one? Answer The example you drew is not a covering space of S1∨S1∨S1, because the unique vertex of S1∨S1∨S1 has valence 6, and … Read more

Showing this function is continuous $f:(x,y)\mapsto x^2+y^2$

I have the following function: $$f:\Bbb R^2 \to \Bbb R,\quad f:(x,y)\mapsto x^2+y^2$$ I want to show that this function is continuous by showing that $f^{-1}((a,b))$ is an open set. How do I approach this? I have proved that the function is a metric already. But I can’t think of how elements are mapped for some … Read more

Existence of a metric space where each open ball is closed and has a limit point

Show that there exists a metric space in which every open ball is closed and contains a limit point. I think that the space {1n∣n∈N,n>0}∪{0} with the standard Euclidean metric is an answer, but it is not true open ball with center 1 is closed. Answer Z with the p-adic metric d(m,n)=p−νp(m−n) has the property … Read more