@@ 8,6 8,7 @@ 
 [topology]: https://en.wikipedia.org/wiki/Topological_space#Definition_via_open_sets
 [proper]: https://en.wikipedia.org/wiki/Proper_map
 [continuous function]: https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces
+[properties quotient space]: https://en.wikipedia.org/wiki/Quotient_space_%28topology%29#Properties
 [Reeb graph]: https://en.wikipedia.org/wiki/Reeb_graph
 [Morse]: https://en.wikipedia.org/wiki/Morse_theory#Fundamental_theorems
 [extended pseudometric]: https://en.wikipedia.org/wiki/Metric_(mathematics)#Extended_metrics

          

@@ 154,7 154,7 @@ such that for all $p \in X$ the estimate
 hold?
 
 Next we may ask, how large $\varepsilon$ needs to be
-in order for the answer to *yes*.
+in order for the answer to be *yes*.
 This is the motivation behind the following
 
 * **Definition.**

          
@@ 175,7 175,7 @@ Next we observe that $M$ satisfies the t
 Let $h \colon Z \rightarrow \R$ be another continuous function,
 then $M(f, h) \leq M(f, g) + M(g, h)$.
 
-Now we recall that we started with two continuous $f$ and $g$
+Now we recall that we started with two continuous functions $f$ and $g$
 and each might describe the luminance of an image
 or the distance to a certain shape
 and this is all good,

          
@@ 282,7 282,7 @@ Completely analogously to the above we h
 Let $h \colon Z \rightarrow \R$ be another continuous function,
 then $\mu(f, h) \leq \mu(f, g) + \mu(g, h)$.
 
-Moreover we have the following relationship to the absolute distance.
+Moreover we have the following relation to the absolute distance.
 
 * **Lemma.**
 We have $\mu(f, g) \leq M(f, g)$.

          
@@ 374,9 374,9 @@ the *category of $\overline{\R}$-spaces*
 
 * **Definition.**
 For two continuous functions $f \colon X \rightarrow \overline{\R}$ and
-$g \colon Y \rightarrow \overline{\R}$
-a *homomorphism $\varphi$ from $f$ to $g$* also denoted by
-$\varphi \colon f \rightarrow g$ is a continuous map
+$g \colon Y \rightarrow \overline{\R}$,
+a *homomorphism $\varphi$ from $f$ to $g$*, also denoted by
+$\varphi \colon f \rightarrow g$, is a continuous map
 $\varphi \colon X \rightarrow Y$ such that the diagram
 $$
 \xymatrix{

          
@@ 396,12 396,12 @@ commutes.
     We define the composition of homomorphisms in the *category
 of $\overline{\R}$-spaces* by the composition of maps.
 
-    The *category of $\R$-spaces* is the full subcategory of
+    The *category of $\R$-spaces* is the [full subcategory][] of
 all real-valued continuous functions.
 
 * *Remark.*
-With these definitions in place a reformulation of question @qAbs
-is whether $f$ and $g$ are isomorphic as objects of the
+With these definitions in place a reformulation of question @qAbs is,
+whether $f$ and $g$ are isomorphic as objects of the
 category of $\overline{\R}$-spaces.
 
 ### Further Prior Art

          

@@ 4,30 4,77 @@ In this section we introduce Reeb graphs
 
 * **Definition** (Reeb Graph)**.**
 Given a continuous function $f \colon X \rightarrow \overline{\R}$
-and $x \in X$
+and $x \in X$,
 let $\pi_f (x)$ be the connected component of $x$ in
 $f^{-1}(f(x))$. In this way we obtain a function
 $\pi_f \colon X \rightarrow \pi_f (X) \subset 2^X$ and we endow
-$\pi_f (X)$ with the quotient topology.
-By the universal property
-of the quotient space there is a unique continuous function
+$\pi_f (X)$ with the quotient topology[^refQuoTop].
+By the
+[universal property of the quotient space][properties quotient space]
+there is a unique continuous function
 $\mathcal{R} f \colon \pi_f (X) \rightarrow \overline{\R}$
 such that
+$$
+\xymatrix{
+  X
+  \ar[r]^{\pi_f}
+  \ar[dr]_f
+  &
+  \pi_f (X)
+  \ar[d]^{\mathcal{R} f}
+  \\
+  &
+  \overline{\R}
+}
+$$
+commutes, i.e.
 $\mathcal{R} f \circ \pi_f = f$.
 
     For another continuous function $g \colon Y \rightarrow \overline{\R}$
 and a homomorphism $\varphi \colon f \rightarrow g$
-we may use the universal property of $\pi_f$ again
+we may use the universal property of $\pi_f$ again,
 to obtain a unique continuous map
 $\mathcal{R} \varphi \colon \pi_f (X) \rightarrow \pi_g (Y)$
-with
+such that
+$$
+\xymatrix{
+  X
+  \ar[d]_{\pi_f}
+  \ar[r]^{\varphi}
+  &
+  Y
+  \ar[d]^{\pi_g}
+  \\
+  \pi_f (X)
+  \ar[r]_{\mathcal{R} \varphi}
+  &
+  \pi_g (Y)
+}
+$$
+commutes, i.e.
 $\mathcal{R} \varphi \circ \pi_f = \pi_g \circ \varphi$.
 
+[^refQuoTop]: see for example [@bredon1993, definition I.13.1]
+
 * **Lemma.**
 Let $f \colon X \rightarrow \overline{\R}$ and
 $g \colon Y \rightarrow \overline{\R}$ be continuous functions
 and let $\varphi \colon f \rightarrow g$ be a homomorphism,
-then $\mathcal{R} g \circ \mathcal{R} \varphi = \mathcal{R} f$.
+then the diagram
+$$
+\xymatrix{
+  \pi_f (X)
+  \ar@/^/[rr]^{\mathcal{R} \varphi}
+  \ar[dr]_{\mathcal{R} f}
+  & &
+  \pi_g (Y)
+  \ar[dl]^{\mathcal{R} g}
+  \\
+  &
+  \overline{\R}
+}
+$$
+commutes.
 Or in other words $\mathcal{R} \varphi$ is a homomorphism
 from $\mathcal{R} f$ to $\mathcal{R} g$
 in the category of $\overline{\R}$-spaces.

          
@@ 101,6 148,6 @@ universal property of $\pi_f$ that
 The previous two lemmata imply that $\mathcal{R}$ is an endofunctor
 on the category of $\overline{\R}$-spaces.
 Later we will define an interleaving distance on Reeb graphs,
-but first we will introduce join trees and their interleavings
-as they are easier to understand and may provide us with some intuition
+but first we will introduce join trees and their interleavings.
+Join trees are easier to understand and may provide us with some intuition
 for understanding the more sophisticated interleavings of Reeb graphs.

          

@@ 46,11 46,11 @@ is shown in the image below.
 Before we get to interleavings we make some auxiliary definitions.
 
 * **Definition.**
-Let
+We set
 $D^{\perp} :=
- \set{(a, b)}{(-\infty, \infty] \times (-\infty, \infty]}{a + b \geq 0}$
-then $D^{\perp}$ is a monoid by component-wise addition.
+ \set{(a, b)}{(-\infty, \infty] \times (-\infty, \infty]}{a + b \geq 0}$.
 
+Component-wise addition yields the structure of a monoid on $D^{\perp}$.
 Next we define two weightings on $D^{\perp}$.
 
 * **Definition.**

          
@@ 150,7 150,7 @@ the *relative interleaving distance of
 
 We note that these definitions of an interleaving distance
 are different from the one by @morozov2013.
-The subsequent corollary shows however that
+However the subsequent corollary shows that
 our absolute interleaving distance is equal to the distance by @morozov2013.
 The reason we gave a different definition is that
 now we can phrase the computation