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# initial topology

Let $X_{i}$, $i\in I$ be any family of topological spaces. We say that a topology $\mathcal{T}$ on $X$ is initial with respect to the family of mappings $f_{i}\colon X\to X_{i}$, $i\in I$, if $\mathcal{T}$ is the coarsest topology on $X$ which makes all $f_{i}$’s continuous.

The initial topology is characterized by the condition that a map $g\colon Y\to X$ is continuous if and only if every $f_{i}\circ g\colon Y\to X_{i}$ is continuous.

Sets $\mathcal{S}=\{f_{i}^{{-1}}(U):U$ is open in $X_{i}\}$ form a subbase for the initial topology, their finite intersections form a base.

E.g. the product topology is initial with respect to the projections and a subspace topology is initial with respect to the embedding.

The initial topology is sometimes called topology generated by a family of mappings [2], weak topology [4] or projective topology. (The term weak topology is used mainly in functional analysis.)

From the viewpoint of category theory, the initial topology is an initial source. (Initial structures, which are a natural generalization of the initial topology, play an important rôle in topological categories and categorical topology.)

# References

- 1
J. Adámek, H. Herrlich, and G. Strecker,
*Abstract and concrete categories*, Wiley, New York, 1990. - 2
R. Engelking,
*General topology*, PWN, Warsaw, 1977. - 3
M. Hušek,
*Categorical topology*, Encyclopedia of General Topology (K. P. Hart, J.-I. Nagata, and J. E. Vaughan, eds.), Elsevier, 2003, pp. 70–71. - 4
S. Willard,
*General topology*, Addison-Wesley, Massachussets, 1970. - 5 Wikipedia’s entry on Initial topology

## Mathematics Subject Classification

54B99*no label found*

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