An exponential mapping over set-valued mappings. (English) Zbl 0999.54013
Summary: The paper presents an approach to “selection homotopy extension” properties of set-valued mappings showing that they become equivalent to usual selection extension properties of exponential set-valued mappings associated to them. As a result, several “controlled” homotopy extension theorems are obtained like consequences of ordinary selection theorems. Also, involving set-valued mappings, a simple proof of the Borsuk homotopy extension theorem is given.
MSC:
| 54C60 | Set-valued maps in general topology |
| 55P05 | Homotopy extension properties, cofibrations in algebraic topology |
| 54C65 | Selections in general topology |
| 54C20 | Extension of maps |
| 54F35 | Higher-dimensional local connectedness |
| 54F45 | Dimension theory in general topology |
| 54C55 | Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) |
| 54B20 | Hyperspaces in general topology |
