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

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

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