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Simply connected space

In [topology] , a [topological space] is called ** simply connected ** (or ** 1-connected ** ) if it is [path-connected] and every path between two points can be continuously transformed, staying within the space, into every other while preserving the two endpoints in question ( [see below for an informal discussion] ).

If a space is not simply connected, it is convenient to measure the extent to which it fails to be simply connected; this is done by the [fundamental group] . Intuitively, the fundamental group measures how the holes behave on a space; if there are no holes, the fundamental group is trivial - equivalently, the space is simply connected.

** Informal discussion **

Informally, a thick object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with handle) is simply connected, but a hollow rubber ball * is * simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are [connected] but not simply connected are called ** nonâ€“simply connected ** or, in a somewhat old-fashioned term, ** multiply connected ** .

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