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On the wiki page on coherent topology, and more precisely, topological union (aka topology generated by a collection of spaces) (http://en.wikipedia.org/wiki/Coherent_topology#Topological_union), it is said that if the generating spaces {X_i} satisfy the compatibility condition that for each i,j, the subspace topologies induced on [itex]X_i\cap X_j[/itex] by X_i and X_j are the same, then the inclusion maps [itex]\iota_i:X_i\rightarrow \cup_iX_i[/itex] are topological embeddings (i.e. homeomorphisms onto their images).

Is this true? I tried proving it but without success but could not find a counter example either.

For instance, CW-complexes are the topological union of their n-skeletons. Is it always true that the inclusion of a n-skeleton into the CW-complex is a topological embedding?

Is this true? I tried proving it but without success but could not find a counter example either.

For instance, CW-complexes are the topological union of their n-skeletons. Is it always true that the inclusion of a n-skeleton into the CW-complex is a topological embedding?

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