engleski

A note on extension theory and direct limits

The direct limit of a direct sequence consisting of normal spaces and closed inclusion mappings is a normal space. This result does not generalize to direct systems because there exist direct systems of compact Hausdorff spaces and closed inclusion mappings whose limit spaces fail to be normal. Introducing additional assumptions, K. Morita obtained normality of the limit space also for systems of normal spaces. Replacing normality of the spaces in the system by the property that a given K (in particular, a CW-complex K) is an absolute extensor for these spaces, the analogue of Morita's theorem remains valid, i.e., K is an absolute extensor for the limit space. This holds even under a weaker version of Morita's additional assumptions. In the case of direct sequences, these weaker conditions are always satisfied and therefore, the improved Morita theorem implies the result for direct sequences of normal spaces.

absolute extensor ; extension dimension ; cohomological dimension ; CW-complex ; direct system ; direct limit ; colimit ; normal space ; compact Hausdorff space ; locally compact space ; compactly generated space ; k-space ; Tychonoff plank ; weak topology.

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