Topology MOC

Proper map

Given topological spaces 𝑋,π‘Œ, a map 𝑓 :𝑋 β†’π‘Œ is called proper iff the preΓ―mage π‘“βˆ’1(𝐾) of every compact 𝐾 βŠ†π‘Œ is compact. topology

Properties

  • If 𝑋 and π‘Œ are locally compact

    • If 𝑋 is second-countable: a continuous map 𝑓 :𝑋 β†’π‘Œ is proper iff every sequence without limit points maps to a sequence without limit points.
    • A continuous map 𝑓 :𝑋 β†’π‘Œ is compact iff the map
    𝑓⋆:π‘‹β‹†β†’π‘Œβ‹†πœ”β†¦πœ”π‘₯βˆˆπ‘‹β†¦π‘“(π‘₯)

    between Alexandroff extensions is continuous.

  • If 𝑋 is compact: all continuous maps are proper, since all compact subsets of π‘Œ are closed and all closed subsets of 𝑋 are compact.

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