We discuss the behaviour of (transfinite) dimension functions, in particular, Cmp and trInd, on finite and countable unions of closed subsets in separable metrizable spaces.
Michael G. CHARALAMBOUS. Vitalij A. CHATYRKO. Yasunao HATTORI. "The behaviour of dimension functions on unions of closed subsets." J. Math. Soc. Japan 56 (2) 489 - 501, April, 2004. https://doi.org/10.2969/jmsj/1191418641