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A planar compactum with connected complement can be an embedded in a cellularcontinuum by attaching a null sequence of arcs.  Two based maps f and g from a planar Peano continuum X to a planar set Y arehomotopic iff f and g induce the same homomorphism between fundamental groups.

Embedding planar compacta in planar continua with application: homotopic  maps between planar Peano continua are characterized by the fundamental group  homomorphism