- #1

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- Summary:
- R without interval and open interval

Hello!

I have two related exercises I need help with

1. Partition the space ##\mathbb{R}## into the interval ##[a,b]##, and singletons disjoint from this interval. The associated equivalence ##\sim## is defined by ##x\sim y## if and only if either##x=y## or ##x,y\in[a,b]##. Then ##\mathbb{R}/\sim## is the space obtained from ##\mathbb{R}## by shrinking ##[a,b]## to a point. The space ##\mathbb{R}/\sim## looks like ##\mathbb{R}## show that it is homeomorphic to ##\mathbb{R}##.

2. Suppose we use the open interval ##(a,b)## in place of ##[a,b]## in the previous excercise. So, in this case ##x\sim y## if and only if either ##x=y## or ##x,y\in (a,b)##. Show that ##\mathbb{R}/\sim## is not homeomorphic to ##\mathbb{R}##.

Concerning the first excercise I have the following

Canonical projection maps initial ##\mathbb{R}## to ##\{ (-\infty,a),point,(b,+\infty)\}##. Here ##point## is the image of the interval ##[a,b]##. It is not had to provide a bijective map with the following properties

$$ f:(-\infty,a)\rightarrow (-\infty,0),\;\; f:point\rightarrow 0,\;\; f:(b,+\infty)\rightarrow (0,+\infty)$$

How to show that this map is continuous based on definition of open sets in quotient topology?

Concerning the second excercise

How to show that something is not homeomorphic? Maybe there is some sort of criterion, cause to show that one can not invent homeomorphism looks like a tough problem.

I have two related exercises I need help with

1. Partition the space ##\mathbb{R}## into the interval ##[a,b]##, and singletons disjoint from this interval. The associated equivalence ##\sim## is defined by ##x\sim y## if and only if either##x=y## or ##x,y\in[a,b]##. Then ##\mathbb{R}/\sim## is the space obtained from ##\mathbb{R}## by shrinking ##[a,b]## to a point. The space ##\mathbb{R}/\sim## looks like ##\mathbb{R}## show that it is homeomorphic to ##\mathbb{R}##.

2. Suppose we use the open interval ##(a,b)## in place of ##[a,b]## in the previous excercise. So, in this case ##x\sim y## if and only if either ##x=y## or ##x,y\in (a,b)##. Show that ##\mathbb{R}/\sim## is not homeomorphic to ##\mathbb{R}##.

Concerning the first excercise I have the following

Canonical projection maps initial ##\mathbb{R}## to ##\{ (-\infty,a),point,(b,+\infty)\}##. Here ##point## is the image of the interval ##[a,b]##. It is not had to provide a bijective map with the following properties

$$ f:(-\infty,a)\rightarrow (-\infty,0),\;\; f:point\rightarrow 0,\;\; f:(b,+\infty)\rightarrow (0,+\infty)$$

How to show that this map is continuous based on definition of open sets in quotient topology?

Concerning the second excercise

How to show that something is not homeomorphic? Maybe there is some sort of criterion, cause to show that one can not invent homeomorphism looks like a tough problem.