absolute extensor



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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topological homotopy theory



A topological space KK is called an absolute extensor if for

  1. XX any nice topological space

  2. AiXA \overset{i}{\hookrightarrow} X any closed subspace

  3. f:AKf\colon A \longrightarrow K any continuous function

there is an extension to a continuous function f˜:XK\tilde{f}:X\to K, i.e., such that f˜=fi\tilde{f}=f\circ i:

A f K i f˜ X \array{ A &\overset{f}{\longrightarrow}& K \\ {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\exists \tilde f}} \\ X }

Here “nice topological space” is variously taken to mean metrizable topological space or at least normal topological space.

A variation of this concept, absolute neighborhood extension, only requires the extension to exist over a neighborhood of AA in XX.


The Tietze extension theorem implies that the real line \mathbb{R} equipped with its Euclidean space metric topology is an absolute extensor. It follows that so are the closed interval subspace [0,1][0,1] \subset \mathbb{R} and the circle S 1S^1.

Products of absolute extensors are absolute extensors, including the Hilbert cube.

The two point discrete space S 0S^0 as well as any sphere S nS^n is an absolute neighborhood extensor, but not an absolute extensor.


Last revised on May 24, 2017 at 02:58:41. See the history of this page for a list of all contributions to it.