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A topological embedding $\gamma :S^1\to X$ of a circle into a topological space. On an oriented surface, an essential such curve is noncontractible; on a punctured surface, nonperipheral means that it does not surround only one puncture.

A continuous map injective map $f:X\to Y$ that is a homeomorphism onto its image with the subspace topology. A continuous map injective map from a compact space into a Hausdorff space is a topological embedding.