The symmetric square of a topological space $X$ is obtained from the usual square $X^2$ by identifying pairs of symmetric points $(x\_1,x\_2)$ and $(x\_2,x\_1)$. Thus, elements of the symmetric square can be identified with unordered pairs $\{x\_1,x\_2\}$ from $X$ (including the degenerate case $x\_1 = x\_2$). With this identification, the following question is very natural:



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> When is there a continuous selector from the symmetric square to the original space?
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This is always possible if $X$ is a generalized ordered space (a subspace of a linear order with the order topology) since min and max are both continuous selectors. I don't think this is a necessary condition since I can construct continuous selectors (on Baire space, for example) that aren't equal to min or max for any ordering of the underlying space.


A necessary condition is that removing the diagonal disconnects $X^2$. More precisely, as Adam Bjorndahl pointed out in the comments, the complement of the diagonal should be the union of two disjoint open sets such that $(x\_1,x\_2)$ is in one iff $(x\_2,x\_1)$ is in the other. So, for example, there are no continuous selectors for the symmetric square of the plane nor the circle.


I'm mostly interested in the case when $X$ is Polish, so it's fine to assume that $X$ is very nice: Hausdorff, normal, perfect, etc.



Continuously selecting elements from unordered pairs
In [this paper](http://www.ams.org/mathscinet-getitem?mr=627702%20) Van Mill and Wattel proved that the existence of a continuous selection characterizes orderability in the class of compact Hausdorff spaces.


