## 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

## Showing the Sum of n−1n-1 Tori is a Double Cover of the Sum of nn Copies of RP2\mathbb{RP}^2

I want to show that the non-orientable surface of genus n has a 2-sheeted cover by an orientable surface of genus n−1. The base cases are easy: S2 covers RP2 and I worked on a proof that the torus T2 covers RP2#RP2≈K (Klein bottle). Here are some of my constraints: Nothing about the relationship between … 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

## Find all covering spaces of RPn×RPn\mathbb{RP}^n \times \mathbb{RP}^n, n>1n>1

Let X=RPn×RPn. I know the following: the universal cover of X is Y=Sn×Sn the fundamental group of X is G=Z/2Z×Z/2Z={(0,0),(0,1),(1,0),(1,1)} Covering spaces of X are defined by actions of subgroups of G on Y. Each of the elements of G generates a subgroup of order two. Clearly the covering spaces defined by the action of … Read more