Two mappings from a CW-complex to a 1-dimensional CW-complex are homotopic if and only if their restrictions to finite subcomplexes are homotopic.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-4-8, author = {Sibe Marde\v si\'c}, title = {There are no Phantom Pairs of Mappings to 1-Dimensional CW-Complexes}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {55}, year = {2007}, pages = {365-371}, zbl = {1139.55013}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-4-8} }
Sibe Mardešić. There are no Phantom Pairs of Mappings to 1-Dimensional CW-Complexes. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) pp. 365-371. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-4-8/