CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
In an (∞,1)-category , for any object its suspension object is the homotopy pushout
where is the terminal object.
This is the mapping cone of the terminal map . See there for more details.
This concept is dual to that of loop space object.
In Top, this is the reduced suspension of a space.
In a category of chain complexes the suspension of a chain complex is given by shifting the degrees of the chain complex up by one.
suspension object