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/