Geodesic
Example of nonnormal submetrizable space of small cardinality
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 2, pp. 180-183

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In 1976 K. Alster and T. Przymusiński constructed an example of nonnormal completely regular separable and first-countable topological space of cardinality $\aleph_1$. Also they proved a theorem which implies that under Martin's Axiom with negation of continuum hypothesis there is no similar example, which is additionally cometrizable. If we weaken the cometrizability condition to submetrizability, then the similar statement can not be proved: here we construct an example of nonnormal completely regular submetrizable separable first-countable locally countable space of cardinality $\aleph_1$.
Keywords: submetrizable space, nonnormal space, cometrizable space. 
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M. A. Patrakeev; E. A. Reznichenko. Example of nonnormal submetrizable space of small cardinality. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 2, pp. 180-183. http://geodesic.mathdoc.fr/item/VUU_2015_25_2_a2/