CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The suspension $S X$ of a topological space $X$ is a space of one higher dimension, which is (for inhabited $X$) a quotient space of $X \times [0,1]$. The difference between $S X$ and $X \times [0,1]$ is that the copy of $X$ at each endpoint ($0$ or $1$) is replaced by a single point.
Compare the reduced suspension $\Sigma X$, where you start with a pointed space and identify all copies of the base point as well. This is the special case in Top of a general operation in (∞,1)-categories: see suspension object. For CW-complexes the reduced suspension is weakly homotopy equivalent to the ordinary suspension: $\Sigma X \simeq S X$.
This picture
from Wikimedia shows the suspension of the blue circle as $X$; the green dots correspond to $2$ in the first definition below.
Note that we replace $X$ by a single point at each endpoint; we don't merely identify all of the points in $X$ there. This only makes a difference for the empty space; we should have $S \empty = \{\bot,\top\}$, not $S \empty = \empty$. (This is related to the issues in the definition of connected space.)
Let $X$ be a space (such as a topological space, or something more interesting like a generalized smooth space). Let $I$ be the unit interval $[0,1]$ in the real line; let $2$ be the $2$-point discrete space $\{\bot,\top\}$. Let $X \times I \times 2$ be the cartesian product of $X$, $2$, and $I$; let $X + (X \times 2 \times I) + 2$ be the disjoint union of $X$, $X \times I \times 2$ and $2$. We will suppress reference to the inclusion maps into $X + (X \times I) + 2$; it will be clear from context how to parse an element of the latter.
The suspension $S X$ of $X$ is the quotient space of $X + (X \times I \times 2) + 2$ by the equivalence relation $\sim$ generated by:
This generalises immediately to an operation called the join $X \star Y$ of two spaces $X$ and $Y$; this is the quotient space of $X + (X \times I \times Y) + Y$ by the equivalence relation generated by:
(Compare join of simplicial sets, the same operation in another guise.) Then we have $S X = X \star 2$.
It is somewhat simpler to define $S X$ as the quotient space of $(X \times I) + 2$ by the equivalence relation generated by:
This works to define a topological space, but it does not directly give the smooth structure that matches the picture above.
If $X$ has a point $p$, hence if it is inhabited, then we can define $S X$ as the quotient space of $X \times I$ by the equivalence relation generated by:
This is probably the most common definition seen, but it only works for $X$ an inhabited space (and even then gives only the topological structure).
To make the suspension of a pointed space $(X,p)$ again a pointed space one may further collapse in $S X$ the set $\{p\} \times I$ to the point. The result then is called the reduced suspension of $(X,p)$ and is denoted
It's easy to extend the suspension operation $S$ to a functor from Top to itself.
For CW-complexes suspension and reduced suspension agree, up to weak homotopy equivalence.
The suspension of the $n$-cube is the $(n+1)$-cube, probably best visualised as a diamond. This gives a recursive definition of cube, starting with the $0$-cube as the point, which is not the suspension of anything. Note that this not only gives us the topological structure of the cube, but also (by working in an appropriate category of smooth spaces throughout) the correct smooth structure on the cube as a manifold with corners. You can probably even get the correct metric on the cube (normalised to have diagonals of length $1$) automatically by using a more complicated quotienting process.
Up to homeomorphism, the suspension of the $n$-sphere is the $(n+1)$-sphere, and the reduced suspension is
See at one-point compactification – Examples – Spheres for details.
Notice that the $n$-sphere is (topologically) the boundary of the $(n+1)$-cube. The coincidence that ‘sphere’ and ‘suspension’ both begin with ‘s’ has not been ignored; we can write $S^n \cong S^n(2)$, where $S^n$ on the left is the $n$-sphere and $S^n$ on the right is the $n$-fold composite of the suspension functor. (Actually, you should start with the $(-1)$-sphere as the empty space, which is not the suspension of anything; then the $0$-sphere is $S \empty = 2$.) However, this does not give the correct smooth structure on the sphere, unless perhaps there is some more sophisticated definition that fixes this (but then that would break the cube). It might be more appropriate to say that the suspension of the $n$-globe is the $(n+1)$-globe.
Up to topological structure, the suspension of the $n$-simplex is the $(n+1)$-simplex, but now this is not very useful. To study simplices, you should use the cone functor instead, which is $\Lambda X = X \star 1$, where $1$ is the point.
Everybody knows about the suspension, but Wikipedia knows about the join. See also the textbook by Hatcher and Postnikov, Homotopy of CW-complexes.
The question on what is the Eckman-Hilton dual to $X\star Y$ find in
Here is Chapter 1 (pdf) of a textbook that knows that $S \empty = 2$, although even it regards this as an exception.
George Whitehead, Some aspects of stable homotopy theory (pdf)
Ralph Cohen, A model for the free loop space of a suspension Lecture Notes in Mathematics, 1987, Volume 1286/1987, 193-207 ([])