A Note on Sequence-covering Images of Metric Spaces

In this brief note, we prove that every space is a sequence-covering image of a topological sum of convergent sequences. As the application of this result, sequence-covering images of locally compact metric spaces (or, locally separable metric spaces, metric spaces) and sequentially-quotient images of a locally compact metric spaces (or, locally separable metric spaces, metric spaces) are equivalent. In [6], Z. Li and Y. Ge proved the following theorem. Theorem 1 ([6], Theorem 6). Let X be a space. Then there exists a metric space M and a pseudo-sequence-covering mapping f : M −→ X. By using this result, the authors obtained that pseudo-sequence-covering images of metric spaces and sequentially-quotient images of metric spaces are equivalent. After that, Y. Ge [4] showed the equivalence of sequence-covering images and sequentiallyquotient images for metric domains as follows. Theorem 2 ([4], Theorem 7). The following are equivalent for a space X. (1) X is a sequence-covering image of a metric space. (2) X is a pseudo-sequence-covering image of a metric space. (3) X is a sequentially-quotient image of a metric space. Take these results into account, note that “sequence-covering =⇒ pseudo-sequencecovering”, and “locally compact metric =⇒ locally separable metric =⇒ metric”, then the following questions are natural. Question 3. Can “pseudo-sequence-covering” in Theorem 1 be replaced by “sequencecovering”? Question 4. Can “metric” in Theorem 2 be replaced by “locally separable metric” or “locally compact metric”? 2000 Mathematics Subject Classification. 54E35, 54E40, 54D80.