2

James Dugundji defines (Topology, chap. VI definition 5.1) a cone in the following manner

For any space $X$, the cone $TX$ over $X$ is the quotient space $(X\times I)/R$, where $R$ is the equivalence relation $(x,1)\sim(x',1)$ for all $x$, $x'\in X$.

My question is on the relation $R$. How is $$R=\{\big((x,1),(x',1)\big)\mid x,x'\in X\}\subset (X\times I)^2$$ an equivalence relation on $X\times I$? It is not necessarily reflexive [$(x,0)\nsim(x,0)$]. What does Dugundji mean with that definition of $R$?

He then, in the next line,writes

Equivalently, $TX=(X\times I)/(X\times 1)$; intuitively, $TX$ is obtained from $X\times I$ by pinching $X\times 1$ to a single point.

And I, again, don't know what he means: how is he taking the quotient space of $X\times I$ with respect to $X\times 1$ which is not a subset of $(X\times I)^2$?

Notation:

  1. Dugundji uses $x=\{x\}$ sometimes (in this case, $x=1$).
  2. $I=[0,1]$.
Sebastián P. Pincheira
  • 2,775
  • 8
  • 25
  • $X \times 1$ does not need to be a subset of $(X \times I)^2$ in order to take a quotient, but rather of $X \times I$, which it is – G. Chiusole Sep 20 '22 at 09:13
  • This makes sense, intuitively. Imagine if $X$ is isomorphic to $S^1$. Then $X\times I$ is a compact cylinder and when we take the quotient we squish the top ring to a point. – John Douma Sep 20 '22 at 16:31

3 Answers3

4

I guess he means the equivalence relation generated by $(x,1)\sim(x',1)$, i.e. the smallest equivalence relation containing all couples $((x,1),(x',1))$. In this case, that means that $(x,a)R(y,b)$ only if $(x,a)=(y,b)$ or $a=b=1$.

In the next line, he considers $X\times\left\{1\right\}$ as a subset of $X\times I$ and he takes the quotient topology of this subset. That is, he lets two points in $X\times I$ be equivalent if and only if they are either equal or both in $X\times\left\{1\right\}$.

Andijvie
  • 1,470
2

A good answer has already been posted. For those following the text Topology by Dugundji (in case useful for someone else stumbling on this question), you have to undertand this definition through the lens of the example 1 of section 4 in chapter VI:

Ex. 1$\quad$For $A\subset X$, let $R_A$ be the equivalence relation $(A\times A)\cup\{(x,x)\mid x\in X\}$. The quotient space $X/R_A$ is the space $X$ with $A$ identified to a point, $[A]$, and is written $X/A$.

This notation diverges from the usual notation of quotient sets present in the text up to that point.

Sebastián P. Pincheira
  • 2,775
  • 8
  • 25
1

What the author means is the equivalence relation induced by $(x,1)∼(x′,1)$. That is, more formally,

$$ \forall ((x_1, y_1),(x_2, y_2)) \in (X \times I)^2: (x_1, y_1) \sim (x_2, y_2) \Leftrightarrow y_1 = y_2 = 1 $$

This is an equivalence relation.

Regarding the other question: as a set, the elements of the quotient space are the equivalence classes of the equivalence relation i.e. subsets of $X \times I$. Then, writing $(X \times I)/(X \times 1)$ denotes the quotient of $X \times I$ by the equivalence relation above.

G. Chiusole
  • 5,594