<p>There are various ways to express the concept of continuity of a binary relation, but one I've come across seems to imply the closed-graph property is sufficient. That is to say:</p>
<p>For a nonempty set <span class="math-container">$X$</span> endowed with a topological structure, a relation <span class="math-container">$R$</span> on <span class="math-container">$X$</span> is (closed) continuous if it is a closed subset of <span class="math-container">$X \times X$</span>.</p>
<p>If <span class="math-container">$X$</span> is a metric space, an equivalent definition is:</p>
<p>If <span class="math-container">$(x_{m}), (y_{m}) \in X^{\infty}$</span> are sequences satisfying <span class="math-container">$x_{m} \rightarrow x$</span>, <span class="math-container">$y_{m} \rightarrow y$</span>, and <span class="math-container">$x_{m} R y_{m}$</span>, for each <span class="math-container">$m$</span>, then <span class="math-container">$x R y$</span>.</p>
<p>Where the implication is that <span class="math-container">$R$</span> contains all its limit points and is thus closed in <span class="math-container">$X \times X$</span>. I've seen this definition applied to preference relations as well, where a preference relation was defined as a binary relation satisfying</p>
<ol>
<li>Reflexivity: For all <span class="math-container">$x \in X$</span>, <span class="math-container">$x R x$</span></li>
<li>Transitivity: For all <span class="math-container">$x, y, z \in X$</span>, if <span class="math-container">$x R y$</span> and <span class="math-container">$y R z$</span> then <span class="math-container">$x R z$</span></li>
<li>Connectedness: For all <span class="math-container">$x, y \in X$</span>, <span class="math-container">$x R y$</span>, <span class="math-container">$y R x$</span>, or both</li>
</ol>
<p>But we can formally define functions and correspondences as binary relations satisfying the total property. That is:</p>
<p>If <span class="math-container">$R$</span> is a self-map/self-correspondence on <span class="math-container">$X$</span> (I use self-map here for consistency with most preference relation setups), then for all <span class="math-container">$x \in X$</span>, there exists a <span class="math-container">$y \in X$</span> such that <span class="math-container">$(x, y) \in R$</span> (equivalently, <span class="math-container">$x R y$</span> or <span class="math-container">$R(x) = y$</span>).</p>
<p>Where functions additionally satisfy the functional property, which specifies that each value in the domain maps to just one value from the range:</p>
<p>If <span class="math-container">$(x, y) \in R$</span> and <span class="math-container">$(x, z) \in R$</span> (equivalently, <span class="math-container">$x R y$</span> and <span class="math-container">$x R z$</span>, or <span class="math-container">$R(x) = y$</span> and <span class="math-container">$R(x) = z$</span>), then <span class="math-container">$y = z$</span>.</p>
<p>In the context of correspondences, we are told that the closed-graph property does not necessarily imply upper hemicontinuity (which reduces to continuity of single-valued functions), unless the range set is additionally compact. But in the above definition of continuity of a binary relation, no reference to the compactness of <span class="math-container">$X$</span> is made. The examples I've seen for correspondences generally referred to singleton-valued correspondences, so I wonder if the functional property is part of the issue.</p>
<p>So my question is, why does the closed-graph property appear to be sufficient for continuity of a binary relation, but not for correspondences or functions? Is it assumed when we discuss binary relations that the functional property does not come into play? Do we generally mean something slightly weaker when we refer to continuity of preference relations?</p>

CC BY-SA 4.0
2024-05-21T19:50:58.627Z
Usually, one calls a correspondence continuous if it is both upper and lower hemicontinuous. I don’t see any relevant connection to continuity of preferences, with some rather esoteric exceptions.
Michael Greinecker
CC BY-SA 4.0
2024-05-21T20:44:02.590Z
Understood! I could have been more clear in the title, but for correspondences I was referring to the proposition that closed-graph alone does not necessarily imply UHC unless the range set (in this case equal to the domain set) is compact. My understanding is UHC reduces to continuity for (single-valued) functions, but wasn't sure whether the above definition of continuity of binary relations was more akin to UHC or UHC+LHC.
hillard28
CC BY-SA 4.0
2024-05-21T20:54:36.160Z
Both UHC and LHC are equivalent to continuity for functions.
Michael Greinecker
CC BY-SA 4.0
2024-05-21T21:03:41.733Z
Yes, sorry, I’m being very loose with my language. Understood that both reduce to continuity of functions. Bit surprised there’s not a tighter relationship across these concepts given functions and correspondences can both be defined in terms of a binary relation on a domain and range set.
hillard28
CC BY-SA 4.0
2024-05-21T21:38:18.310Z
There is in many settings (second countable topological spaces) a very close connection between continuity of preferences and functions: A preference relation is continuous if and only if it can be represented by a continuous utility function. I guess this is where the name comes from.
Michael Greinecker
<microeconomics><mathematical-economics><preferences><proof>
Reconciling Continuity of Binary/Preference Relations with Continuity of Functions/Correspondences
hillard28
# Reconciling Continuity of Binary/Preference Relations with Continuity of Functions/Correspondences

(Asked by: username_0 on 2024-05-21)

There are various ways to express the concept of continuity of a binary relation, but one I've come across seems to imply the closed-graph property is sufficient. That is to say:

For a nonempty set $X$ endowed with a topological structure, a relation $R$ on $X$ is (closed) continuous if it is a closed subset of $X \times X$.

If $X$ is a metric space, an equivalent definition is:

If $(x\_{m}), (y\_{m}) \in X^{\infty}$ are sequences satisfying $x\_{m} \rightarrow x$, $y\_{m} \rightarrow y$, and $x\_{m} R y\_{m}$, for each $m$, then $x R y$.

Where the implication is that $R$ contains all its limit points and is thus closed in $X \times X$. I've seen this definition applied to preference relations as well, where a preference relation was defined as a binary relation satisfying

1. Reflexivity: For all $x \in X$, $x R x$
2. Transitivity: For all $x, y, z \in X$, if $x R y$ and $y R z$ then $x R z$
3. Connectedness: For all $x, y \in X$, $x R y$, $y R x$, or both

But we can formally define functions and correspondences as binary relations satisfying the total property. That is:

If $R$ is a self-map/self-correspondence on $X$ (I use self-map here for consistency with most preference relation setups), then for all $x \in X$, there exists a $y \in X$ such that $(x, y) \in R$ (equivalently, $x R y$ or $R(x) = y$).

Where functions additionally satisfy the functional property, which specifies that each value in the domain maps to just one value from the range:

If $(x, y) \in R$ and $(x, z) \in R$ (equivalently, $x R y$ and $x R z$, or $R(x) = y$ and $R(x) = z$), then $y = z$.

In the context of correspondences, we are told that the closed-graph property does not necessarily imply upper hemicontinuity (which reduces to continuity of single-valued functions), unless the range set is additionally compact. But in the above definition of continuity of a binary relation, no reference to the compactness of $X$ is made. The examples I've seen for correspondences generally referred to singleton-valued correspondences, so I wonder if the functional property is part of the issue.

So my question is, why does the closed-graph property appear to be sufficient for continuity of a binary relation, but not for correspondences or functions? Is it assumed when we discuss binary relations that the functional property does not come into play? Do we generally mean something slightly weaker when we refer to continuity of preference relations?

**Tags:** microeconomics, mathematical-economics, preferences, proof

> **Comments:**
> **username_1**: Usually, one calls a correspondence continuous if it is both upper and lower hemicontinuous. I don’t see any relevant connection to continuity of preferences, with some rather esoteric exceptions.  
> **username_0**: Understood! I could have been more clear in the title, but for correspondences I was referring to the proposition that closed-graph alone does not necessarily imply UHC unless the range set (in this case equal to the domain set) is compact. My understanding is UHC reduces to continuity for (single-valued) functions, but wasn't sure whether the above definition of continuity of binary relations was more akin to UHC or UHC+LHC.  
> **username_1**: Both UHC and LHC are equivalent to continuity for functions.  
> **username_0**: Yes, sorry, I’m being very loose with my language. Understood that both reduce to continuity of functions. Bit surprised there’s not a tighter relationship across these concepts given functions and correspondences can both be defined in terms of a binary relation on a domain and range set.  
> **username_1**: There is in many settings (second countable topological spaces) a very close connection between continuity of preferences and functions: A preference relation is continuous if and only if it can be represented by a continuous utility function. I guess this is where the name comes from.  

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## 0 answers

