<p>For a Microeconomics Course, we are going through MWG, and in the lecture we discussed the notion of a continuous preference relation.
A preference relation <span class="math-container">$\succsim$</span> on a set <span class="math-container">$X$</span> is called continuous if
<span class="math-container">$\forall x,y\in X$</span>, for all sequences <span class="math-container">$\{x_k\}\to x,\{y_k\}\to y$</span> for which we <span class="math-container">$x_k \succsim y_k,\forall k$</span>, we have that <span class="math-container">$x \succsim y$</span>. I am trying to understand the choice of words for continuity of the preference relation here: by some steps, using a theorem from Debreu, we can get from a rational and continuous preference relation to a continuous utility function that represents it. However, can we phrase the above definition in such a way that it corresponds to the usual topological notion of continuity? (A function is continuous if every pre-image of an open set is open.)</p>
<p>What spaces and what function should we consider, to find an equivalent topological definition of continuity of a preference relation? I got as far that it must be either some map <span class="math-container">$f: X\times X \to X$</span>, or a map <span class="math-container">$f: X\times X\to \{0,1\}$</span>, with X in the topology induced by <span class="math-container">$\succsim$</span> and <span class="math-container">$\{0,1\}$</span> in the discrete topology. But where to from here?</p>

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2020-09-03T19:34:23.453Z
An equivalent statement is "$\succsim$ on a topological space $X$ is continuous if $ \succsim_y  = \{x | x \succsim y\}$ is closed for all $y$". As usual, replace sequences by nets if topology is not metrizable.  The set $\succsim_y$ is the preimage of $y$ if the binary relation $\succsim$ is a function, which recovers the definition for functions.
Michael
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2020-09-03T20:49:26.660Z
@Michael: Shouldn't the lower contour set $\precsim_y=\{x|y\succsim x\}$ be closed as well?
Herr K.
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2020-09-04T07:12:34.333Z
@Michael, thanks, that makes sense indeed, I forgot that we could look at it like that. Herr K., you are correct, but that sounds like an immediate consequence of the fact that either of $\succsim$ and $\precsim$ as mappings are continuous iff the other is, so I am not sure if the requirement is necessary or whether it immediately follows from the condition stated by Michael.
J. Dekker
CC BY-SA 4.0
2020-09-04T07:39:00.193Z
@J.Dekker Consider the preference relation $\succeq$ on $\mathbb{R}$ given by $x\succeq y$ if and only if $\lfloor x \rfloor\geq\lfloor y \rfloor$, where $\lfloor r \rfloor$ is the greatest integer smaller or equal to $r$. The preference relation $\succeq$ has closed upper contour sets but not all lower contour sets are closed.
Michael Greinecker
CC BY-SA 4.0
2020-09-04T09:52:04.573Z
As everyone correctly points out, closedness of $\precsim_y$'s should be included. (Similarly, for real-valued function $f$, closedness of the sets $\{ f \geq y\}_y$ only gives upper semicontinuity---e.g. the integer part function in @MichaelGreinecker's comment.)
Michael
CC BY-SA 4.0
2020-09-05T08:15:23.587Z
@MichaelGreinecker thank you for pointing it out, I see the logical flaw in my argument now
J. Dekker
<microeconomics><mathematical-economics><real-analysis>
Topological intuition to continuous preference relation
J. Dekker
# Topological intuition to continuous preference relation

(Asked by: username_0 on 2020-09-03)

For a Microeconomics Course, we are going through MWG, and in the lecture we discussed the notion of a continuous preference relation.
A preference relation $\succsim$ on a set $X$ is called continuous if
$\forall x,y\in X$, for all sequences $\{x\_k\}\to x,\{y\_k\}\to y$ for which we $x\_k \succsim y\_k,\forall k$, we have that $x \succsim y$. I am trying to understand the choice of words for continuity of the preference relation here: by some steps, using a theorem from Debreu, we can get from a rational and continuous preference relation to a continuous utility function that represents it. However, can we phrase the above definition in such a way that it corresponds to the usual topological notion of continuity? (A function is continuous if every pre-image of an open set is open.)

What spaces and what function should we consider, to find an equivalent topological definition of continuity of a preference relation? I got as far that it must be either some map $f: X\times X \to X$, or a map $f: X\times X\to \{0,1\}$, with X in the topology induced by $\succsim$ and $\{0,1\}$ in the discrete topology. But where to from here?

**Tags:** microeconomics, mathematical-economics, real-analysis

> **Comments:**
> **username_1**: An equivalent statement is "$\succsim$ on a topological space $X$ is continuous if $ \succsim_y  = \{x | x \succsim y\}$ is closed for all $y$". As usual, replace sequences by nets if topology is not metrizable.  The set $\succsim_y$ is the preimage of $y$ if the binary relation $\succsim$ is a function, which recovers the definition for functions.  
> **username_2**: @username_1: Shouldn't the lower contour set $\precsim_y=\{x|y\succsim x\}$ be closed as well?  
> **username_0**: @username_1, thanks, that makes sense indeed, I forgot that we could look at it like that. username_2., you are correct, but that sounds like an immediate consequence of the fact that either of $\succsim$ and $\precsim$ as mappings are continuous iff the other is, so I am not sure if the requirement is necessary or whether it immediately follows from the condition stated by username_1.  
> **username_3**: @username_0 Consider the preference relation $\succeq$ on $\mathbb{R}$ given by $x\succeq y$ if and only if $\lfloor x \rfloor\geq\lfloor y \rfloor$, where $\lfloor r \rfloor$ is the greatest integer smaller or equal to $r$. The preference relation $\succeq$ has closed upper contour sets but not all lower contour sets are closed.  
> **username_1**: As everyone correctly points out, closedness of $\precsim_y$'s should be included. (Similarly, for real-valued function $f$, closedness of the sets $\{ f \geq y\}_y$ only gives upper semicontinuity---e.g. the integer part function in @username_3's comment.)  
> **username_0**: @username_3 thank you for pointing it out, I see the logical flaw in my argument now  

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