# 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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