Universal and proximately universal limits

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We present sufficient conditions on an approximate mapping F : H --> Y of approximate inverse systems in order that the limit f : X --> Y of F is a universal map in the sense of Holsztynski. A similar theorem holds for a more restrictive concept of a proximately universal map introduced recently by the second author. We get as corollaries some sufficient conditions on an approximate inverse system implying that the its limit has the (proximate) fixed point property. In particular, every chainable compact Hausdorff space has the proximate fixed point property.
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