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'initial topology'
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| Title of object: |
initial topology |
| Canonical Name: |
InitialTopology |
| Type: |
Definition |
| Created on: |
2005-09-10 11:54:47 |
| Modified on: |
2005-09-14 04:11:08 |
| Classification: |
msc:54B99 |
| Keywords: |
coarser |
Preamble:
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Content:
\section*{Initial topology}
We say that a topology $\mathcal T$ on $X$ is initial with respect to the family of mappings $\Map {f_i}X{X_i}$, $i\in I$, 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 $\Map gYX$ is continuous if and only if every $\Map {f_i \circ g}Y{X_i}$ is continuous.
Sets $\mathcal S=\{f^{-1}(U): U$ 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 \cite{engelking}, weak topology \cite{willard} 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 initial topology, play an important r\^{o}le in topological categories and categorical topology.)
\begin{thebibliography}{9}
\bibitem{ahs}
J.~Ad\'amek, H.~Herrlich, and G.~Strecker, \emph{Abstract and concrete categories}, Wiley,
New York, 1990.
\bibitem{engelking}
R.~Engelking, \emph{General topology}, PWN, Warsaw, 1977.
\bibitem{husek}
M.~Hu\v{s}ek, \emph{Categorical topology}, Encyclopedia of General Topology (K.~P. Hart,
J.-I. Nagata, and J.~E. Vaughan, eds.), Elsevier, 2003, pp.~70--71.
\bibitem{willard}
S.~Willard, \emph{General topology}, Addison-Wesley, Massachussets, 1970.
\bibitem{wikiinit} Wikipedia's entry on \PMlinkexternal{Initial topology}{http://en.wikipedia.org/wiki/Initial_topology}
\end{thebibliography} |
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