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I am reading a paper by Carlsson on "Topological pattern recognition for point cloud data." I was having a little trouble understanding the formal definition of an equivalence relation and equivalence class in relation to persistent homology. Could someone please clarify the definition below. I have my own interpretation--listed below, but please correct me if I am wrong.

From the text:

A (binary) relation on a set $X$ is a subset of $X × X$. We will often denote relations by $\sim$, and write $x ∼ x$ to indicate that $(x, x)$ is in the relation.

Definition 2.1. A relation ∼ on a set X is an equivalence relation if the following three conditions hold:

  1. $x ∼ x$ for all $x \in X$,
  2. $x ∼ x'$ if and only if $x' ∼ x$,
  3. $x ∼ x'$ and $x' ∼ x''$ implies $x ∼ x''$

By the equivalence class of $x \in X$, denoted by $[x]$, we will mean the set $$\{x | x ∼ x \}$$

My interpretation

So as I interpret it, $\forall x \in X$, then $X \times X := \{(x,x)| x \in X\}$. If we denote $A = \{(x,x)| x \in X\}$, then there exists an equivalence relation between $x_1 \sim x_2$ if

  1. both $ x_1, x_2 \in X$
  2. $(x_1, x_2) \in A$
  3. $(x_2, x_1) \in A$
  4. if $(x_1, x_2) \in A$ and $(x_2, x3) \in A$, then $(x_1, x_3) \in A$.

I think this is the right interpretation, but just wanted to make sure I was on the right track.

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in persistent homology, two cycles are said to belong to the same homology class if their difference is a boundary.

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