Star-like implies contractible

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., topologically star-like space) must also satisfy the second topological space property (i.e., contractible space)
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Statement

Any star-like subset of Euclidean space is a contractible space. In fact, it has a contracting homotopy to any point in the kernel that has the following additional properties:

• The homotopy is a linear homotopy.
• The homotopy is a semi-sudden homotopy, i.e., for $t<1$ the map $x\mapsto F(x,t)$ is a homeomorphism to its image.
• The homotopy is a (strong) deformation retraction.

In particular, a star-like subset of Euclidean space (and more generally, a topologically star-like space) is a semi-suddenly contractible space as well as a SDR-contractible space.

Related facts

• Star-like implies any retraction into the kernel is a strong deformation retraction