Simply Connected Spaces

Topology · Axiom Academy

LESSON Simply Connected Spaces Understanding when a space has no holes In topology, a space is called simply connected if it satisfies two key properties: it must be path-connected, and every loop can be continuously shrunk to a point. A topological space is simply connected if: The fundamental group is trivial: The fundamental group captures the idea of loops in a space. When this group is trivial (contains only the identity element), we say the space is simply connected. For a path-connected space with basepoint , the fundamental group consists of equivalence classes of loops based at , where two loops are equivalent if one can be continuously deformed into the other. Step 3: Example - Euclidean Space The most fundamental example of a simply connected space is Euclidean space . Any loop in can be continuously shrunk to a point. In , there are no obstructions - we can imagine "pulling" any loop tighter and tighter until it becomes a single point. This works for any . Spheres behave differently depending on their dimension. The circle is not simply connected, but higher-dimensional spheres for are simply connected! On a 2-sphere (ordinary sphere), any loop can slide over the surface and contract to a point. The key is that we can move "around" any potential obstacle in the extra dimension. Step 5: Non-Example - The Circle

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