If X is locally compact, then it has this Heine-Borel-property. For topological vector spaces locally compactness is equivalent to finite dimension if I remember correctly.
But there are other examples even vector spaces that have the Heine-Borel-property without being locally compact. The space H(U) of holomorphic functions on an open set UsubseteqmathbbC with the topology of locally uniform convergence of all derivatives. This is Montel's theorem and therefore such spaces are called Montel spaces (well a certain additional condition is needed, but that's not the point) Another example is the Schwartz-Space mathcalS(mathbbRn) of rapidly decreasing functions. Because being Montel is stable under taking strong duals, the space of tempered distributions mathcalS′(mathbbRn) has the Heine-Borel-property too (but is not metrizable).
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