Weakly contractible not implies contractible
This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property need not satisfy the second topological space property
View a complete list of topological space property non-implications | View a complete list of topological space property implications |Get help on looking up topological space property implications/non-implications
|
Statement
It is possible for a topological space to be a weakly contractible space but not a contractible space.
Definitions used
Weakly contractible space
Further information: weakly contractible space
A topological space is termed weakly contractible if it is a path-connected space and all its homotopy groups are trivial.
Contractible space
Further information: contractible space
A topological space is termed contractible if it is homotopy-equivalent to a point.
Proof
Further information: double comb space
The double comb space is an exampe of a weakly contractible space that is not contractible. This is a subset of the Euclidean plane equipped with the subspace topology. Fill this in later