CW implies paracompact

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This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property must also satisfy the second topological space property
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This article involves a proof using cellular induction, viz, it inductive construction on the n-skeleton of a cellular space

Statement

Any CW-space (viz a space that admits a CW-complex structure) is paracompact -- viz every open cover has a locally finite open refinement.

References

  • Topology of CW complexes by A. T. Lundell and S. Weingram, P. 54