Homotopic continuous maps are maps that can be continuously deformed into one another. The corresponding notion for
topological spaces is homotopy
equivalence. For example, the real line and any
tree are homotopy equivalent
to a single point. The circle
is not contractible, but is homotopy
equivalent to a solid torus. Two continuous
maps 
and 
are homotopic if there is
a continuous map
![F:X×[0,1]->Y](/images/equations/Homotopic/NumberedEquation1.svg) |
such that 
and 
.
Whether an inclusion map is homotopic to a constant function depends on the ambient space. For example,
the inclusion of the unit circle into the plane
is homotopic to a constant function, whereas
its inclusion into the punctured plane 
is not. The puncture can be thought
of as an obstacle.
However, there is a way to compare two topological spaces via homotopy without ambient spaces.
Two topological spaces 
and 
are homotopy equivalent if there are continuous
maps 
and 
such that the composition 
is homotopic to the identity
map of 
and 
is homotopic to the identity
map of 
.
For example, the circle is not homotopy
equivalent to a point, for then a constant
function would be homotopic to the identity map
of a circle, which is impossible because they have different
Brouwer degrees.
