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Difference between revisions of "Pre-topological space"

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(Start article: Pre-topological space)
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#REDIRECT [[Closure space]]
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{{TEX|done}}{{MSC|54A05}}
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Let $X$ be a set and $\mathcal{P}X$ the set of subsets of $X$.  A pre-topological space structure on $X$ is defined by a ''Čech closure operator'', a mapping $C : \mathcal{P}X \rightarrow \mathcal{P}X$ such that
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C1) $C(\emptyset) = \emptyset$;
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C2) $A \subseteq C(A)$;
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C3) $C(A \cup B) = C(A) \cup C(B)$.
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A set $A$ in $X$ is ''closed'' if $A = C(A)$.
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A mapping between pre-topological spaces $f : X \rightarrow Y$ is ''continuous'' if $f(C_X(B)) \subseteq C_Y(f(B))$ for any $B \subseteq C$.
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If the operator $C$ also satisfies (C4) $C(C(A)) = C(A)$, then $X$ is a [[topological space]] with $C$ as the [[Kuratowski closure operator]]. 
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==References==
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N.M. Martin,  S. Pollard,  "Closure spaces and logic" , Kluwer Acad. Publ.  (1996)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Graduate Texts in Mathematics '''27''' Springer  (1975) ISBN 0-387-90125-6  {{ZBL|0306.54002}}</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  D. Dikranjan,  W. Tholin,  "Categorical structures of closure operators" , Kluwer Acad. Publ.  (1996)</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  Jürgen Jost, "Mathematical Concepts", Springer (2015) ISBN 331920436X</TD></TR>
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</table>

Revision as of 20:05, 19 January 2021

2020 Mathematics Subject Classification: Primary: 54A05 [MSN][ZBL]

Let $X$ be a set and $\mathcal{P}X$ the set of subsets of $X$. A pre-topological space structure on $X$ is defined by a Čech closure operator, a mapping $C : \mathcal{P}X \rightarrow \mathcal{P}X$ such that

C1) $C(\emptyset) = \emptyset$;

C2) $A \subseteq C(A)$;

C3) $C(A \cup B) = C(A) \cup C(B)$.

A set $A$ in $X$ is closed if $A = C(A)$.

A mapping between pre-topological spaces $f : X \rightarrow Y$ is continuous if $f(C_X(B)) \subseteq C_Y(f(B))$ for any $B \subseteq C$.

If the operator $C$ also satisfies (C4) $C(C(A)) = C(A)$, then $X$ is a topological space with $C$ as the Kuratowski closure operator.

References

[1] N.M. Martin, S. Pollard, "Closure spaces and logic" , Kluwer Acad. Publ. (1996)
[2] J.L. Kelley, "General topology" , Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002
[3] D. Dikranjan, W. Tholin, "Categorical structures of closure operators" , Kluwer Acad. Publ. (1996)
[4] Jürgen Jost, "Mathematical Concepts", Springer (2015) ISBN 331920436X
How to Cite This Entry:
Pre-topological space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-topological_space&oldid=51439