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# Contents

## Definition

A $\Delta$-generated space is a topological space $X$ whose topology is the final topology induced by all maps $\Delta^n \to X$, where $\Delta^n$ runs over all the standard simplices.

Equivalently, the class of $\Delta$-generated spaces is the closure of $\Delta^n$ under small colimits in topological spaces.

## Properties

The category of $\Delta$-generated spaces is coreflective in Top. It is also locally presentable and cartesian closed, and supports a model structure. Thus, it is a nice category of spaces.

## References

$\Delta$-generated spaces were originally proposed by Jeff Smith as a nice category of spaces for homotopy theory. A proof that the category of $\Delta$-generated spaces is locally presentable is in: