nLab absolute extensor

Idea

A topological space $K$ is called an absolute extensor if for

1. $X$ any nice topological space

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

3. $f\colon A \longrightarrow K$ any continuous function

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

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

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 $A$ in $X$.

Examples

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 $[0,1] \subset \mathbb{R}$ and the circle $S^1$.

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

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

Related concepts

• Tietze extension theorem

• quasi-finite CW-complex

• absolute retract

References

• eom, Absolute neighbourhood extensor