# Combining spaces

Another way to construct spaces is by combining them in various ways, as described in the following table. We denote the unit interval $${[0,1]}$$ by $${I}$$. $${X-Y}$$ simply denotes the usual removal of a subset $${Y}$$.

OperationExample
Product: $${X\times Y=}$$ all points $${\left(x,y\right)}$$ with $${x\in X}$$, $${y\in Y}$$
Example: $${S^{1}\times S^{1}=T^{2}}$$, the torus
Wedge sum: $${X\vee Y}$$ identifies a point from each space
Example: $${S^{1}\vee S^{1}=}$$ the figure eight
Quotient: $${X/A}$$ collapses $${A\subset X}$$ to a point
Example: $${S^{2}/S^{1}=S^{2}\vee S^{2}}$$
Suspension: $${SX=}$$ the quotient of $${X\times I}$$ with $${X\times\left\{ 0\right\} }$$ and $${X\times\left\{ 1\right\} }$$ collapsed to points
Example: $${SS^{1}=S^{2}}$$
Join: $${X*Y=}$$ all line segments from $${X}$$ to $${Y}$$; more precisely, $${X\times Y\times I}$$ with $${X\times Y\times\left\{ 0\right\} }$$ collapsed to $${X}$$, $${X\times Y\times\left\{ 1\right\} }$$ to $${Y}$$
Example: $${I*I=}$$ solid tetrahedron

In the case of two disjoint connected $${n}$$-manifolds $${X}$$ and $${Y}$$, we can also define the connected sum $${X\#Y}$$, obtained by removing the interiors of closed $${n}$$-balls from each and identifying the resulting boundary spheres.

The above depicts the connected sum $${S^{2}\#S^{2}=S^{2}}$$.

Some facts about combining spheres are:

• If the product $${X\times Y=S^{n}}$$ then one of the spaces is a point
• The quotient $${S^{n}/S^{n-1}=S^{n}\vee S^{n}}$$ yields a wedge sum
• The suspension $${SS^{n}=S^{n+1}}$$
• The join $${S^{n}*S^{m}=S^{m+n+1}}$$
• The connected sum of $${n}$$-dimensional manifolds $${M\#S^{n}=M}$$