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In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected [1]) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no ...
Since a simply connected space is, by definition, also required to be path connected, any simply connected space is also connected. If the "path connectedness" requirement is dropped from the definition of simple connectivity, a simply connected space does not need to be connected.
The Hawaiian earring is a space which is neither locally simply connected nor simply connected. The cone on the Hawaiian earring is contractible and therefore simply connected, but still not locally simply connected. All topological manifolds and CW complexes are locally simply connected. In fact, these satisfy the much stronger property of ...
The Hawaiian earring is not semi-locally simply connected.. A simple example of a space that is not semi-locally simply connected is the Hawaiian earring: the union of the circles in the Euclidean plane with centers (1/n, 0) and radii 1/n, for n a natural number.
Terms involving connected are also used for properties that are related to, but clearly different from, connectedness. For example, a path-connected topological space is simply connected if each loop (path from a point to itself) in it is contractible; that is, intuitively, if there is essentially only one way to get from any point to any other ...
In general, a space contains a 1-dimensional-boundary hole if and only if it is not simply-connected. Hence, simply-connected is equivalent to 1-connected. X is 0-connected but not 1-connected, so =. The lowest dimension of a hole is 2, so =.
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Namely, a simply connected CW complex is a rational space if and only if its homology groups (,) are rational vector spaces for all >. [5] The rationalization of a simply connected CW complex X {\displaystyle X} is the unique rational space X → X Q {\displaystyle X\to X_{\mathbb {Q} }} (up to homotopy equivalence) with a map X → X Q ...