Connectedness is continuous image-closed
From Topospaces
This article gives the statement, and possibly proof, of a topological space property (i.e., connected space) satisfying a topological space metaproperty (i.e., continuous image-closed property of topological spaces)
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Contents
Statement
Suppose is a connected space and
is a continuous map. Then,
, endowed with the subspace topology from
, is a connected space.
Related facts
Weaker facts
Applications
- Intermediate-value theorem: This states that any continuous real-valued function on a connected space that takes the real values
and
with
, takes all real values in the interval
.
Proof
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