# Tagged Questions

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

3answers
174 views

### Are strongly equivalent metrics mutually complete?

Maybe I'm missing something, but I can't seem to find any references to my exact question. If two metrics, $d_1(x,y)$ and $d_2(x,y)$ are strongly equivalent, then there exists two positive constants, ...
0answers
39 views

### Coconvergent topology basis?

Consider the space $X=\{\frac1n:n\in\mathbb N_+\}$, with the "coconvergent topology": $$\mathcal O=\{A:(A=\varnothing)\lor(\sum_{x\notin A}x<\infty)\}$$ That is, a nonempty set is open iff its ...
2answers
52 views

### Show that the following mapping is a contraction.

I have the following problem from a past paper: "Show that the mapping, $$T(x_1,x_2)=\left(\frac{x_1+2x_2}5-1,\frac{x_1-2x_2}7+1\right)$$ is a contraction on $(\mathbb R^2,d_\infty)$." I ...
1answer
44 views

### How to prove that space is not connected

I found a definition that the space $M$ is not connected if there are open subsets $A,B$ such that $M=A\cup M,A\ne\emptyset\ne B,$ and $A\cap B=\emptyset$. How can I prove from the definition that ...
0answers
58 views

### What theorems or frameworks explain why the roots of well-behaved functions $h : \mathbb{R} \leftarrow \mathbb{R}^2$ seem to be made up of “pieces”?

First, some terminology: given functions $g,f:Y \leftarrow X$, the equalizer of $g$ and $f$ is defined to be the set of all solutions $x \in X$ to the equation $g(x)=f(x)$ in $Y$. Okay. The following ...
2answers
68 views

### How to determine whether those sets are open or closed?

Given those three sets below, A (left), B (center) and C (right), with A, B, C $\subseteq \mathbb{R^2}$, how can I determine, whether they are open or closed in metric space terminology via simplest ...
0answers
54 views

### QFT and topology

I have had a course in topology, I have heard of homotopy quantum field theory and topological field theory, but I dont know anything about QFT, what would be a good starting point to learn about the ...
2answers
40 views

2answers
72 views