Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. Such that there is an interval of maps joining one to the other.
from The Century Dictionary and Cyclopedia
 Occurring at the same place in the offspring as in the parent.
Etymologies
Sorry, no etymologies found.
Examples

If a pair of nuclei can be interchanged by rotation about an axis of symmetry of the molecule then they are chemically equivalent and are called homotopic. e.g. the pair of protons in dichloromethane are chemically equivalent.

Let us pass to another analogy for now: a particle moves around in a topological space, and we wish to describe its path as simply as possible, and such that two paths that may be deformed into each other (i.e., are homotopic), are given the same description.

To prove carefully that paths going around the circle different numbers of times are not homotopic to each other requires a bit more machinery than is developed in this article.

A space is termed simply connected if every loop in the space is homotopic to a constant loop.

Two paths γ0, γ1 are said to be homotopic if there is a continuous function

Two paths are said to be homotopic if one can be continuously deformed/moved/stretched into the other.

In a simply connected space, all loops are homotopic and thus represent a single homotopy class, and so the fundamental group is the trivial group, with only one element.

This stretching can't pass over a hole, so a characterization of homotopic paths yields a characterization of holes. topological space X, the fundamental group is an associated algebraic object which describes the paths that are not homotopic, and hence describes the set of "holes" in the space.

Let us pass to another analogy for now: a particle moves around in a topological space, and we wish to describe its path as simply as possible, in such a way that two paths that may be deformed into each other (i.e., are homotopic), are given the same description.

It's difficult to phrase this elegantly: the fundamental group isn't the set of all loops with the property that all are homotopic.
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