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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?

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    $\begingroup$ 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. $\endgroup$ – Michael Sep 3 at 19:34
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    $\begingroup$ @Michael: Shouldn't the lower contour set $\precsim_y=\{x|y\succsim x\}$ be closed as well? $\endgroup$ – Herr K. Sep 3 at 20:49
  • $\begingroup$ @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. $\endgroup$ – J. Dekker Sep 4 at 7:12
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    $\begingroup$ @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. $\endgroup$ – Michael Greinecker Sep 4 at 7:39
  • $\begingroup$ 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.) $\endgroup$ – Michael Sep 4 at 9:52

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