Let Y be a topological space. Suppose is a continuous map such that for all , and all . (Here v0 denotes the vector v with 0 appended, etc.) Then the image of f is path connected.
Let M be a topological space. Give the product topology and let the symmetric group act on by permuting the coordinates. The space , which parameterizes n-element multisets, can be given the quotient topology.
If is connected, and the multiset is in A for some , then the subset of all coordinates of points in A is connected.