PlanetMath: topological space

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topological space

(Definition)

A topological space is a set together with a set $\mathcal{T}$ whose elements are subsets of , such that

$\emptyset \in \mathcal{T}$
$X \in \mathcal{T}$
If $U_j \in \mathcal{T}$ for all $j \in J$ , then $\bigcup_{j \in J} U_j \in \mathcal{T}$
If $U \in \mathcal{T}$ and $V \in \mathcal{T}$ , then $U \cap V \in \mathcal{T}$

Elements of $\mathcal{T}$ are called open sets of . The set $\mathcal{T}$ is called a topology on . A subset $C \subset X$ is called a closed set if the complement $X \setminus C$ is an open set.

A topology $\mathcal{T}'$ is said to be finer (respectively, coarser) than $\mathcal{T}$ if $\mathcal{T}' \supset \mathcal{T}$ (respectively, $\mathcal{T}' \subset \mathcal{T}$ ).

Examples

The discrete topology is the topology $\mathcal{T} = \mathcal{P}(X)$ on , where $\mathcal{P}(X)$ denotes the power set of . This is the largest, or finest, possible topology on .
The indiscrete topology is the topology $\mathcal{T} = \{\emptyset, X\}$ . It is the smallest or coarsest possible topology on .
Subspace topology
Product topology
Metric topology

Bibliography

1: J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
2: J. Munkres, Topology (2nd edition), Prentice Hall, 1999.

"topological space" is owned by djao. [ full author list (2) ]

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Other names:

topology

Also defines:

open, closed

Keywords:

open set, closed set

Attachments:: cofinite and cocountable topologies (Definition) by yark
Sorgenfrey line (Example) by yark
Niemytzki plane (Example) by PrimeFan
some structures on $\mathbb{R}^n$ (Definition) by drini
list of common topologies (Topic) by matte

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Cross-references: metric topology, product topology, subspace topology, indiscrete topology, power set, discrete topology, coarser, finer, complement, closed set, subsets
There are 676 references to this entry.

This is version 7 of topological space, born on 2001-10-19, modified 2005-08-13.
Object id is 380, canonical name is TopologicalSpace.
Accessed 51927 times total.

Classification:

AMS MSC:	54-00 (General topology :: General reference works )
	55-00 (Algebraic topology :: General reference works )
	22-00 (Topological groups, Lie groups :: General reference works )

Pending Errata and Addenda

None.[ View all 2 ]

Discussion

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topologies on complete lattices by Schneemann on 2006-02-18 13:56:14

Can a topology be defined as a subset of an arbitrary complete (and complemented) lattice, instead of a power set?
If yes, is this lattice required to be complemented? Does it have to be a distributed lattice?

A definition would look like that (tex code). \bigwedge is the infimum operator, \bigvee the supremum operator induced by the non-reflexive order relation "<".

\begin{defi}[topology]
Let $(L, <)$ be a complete lattice. A set $T \subseteq L$ is a topology on $L$, if
\begin{itemize}
\item[(i)] $\bigwedge L, \bigvee L \in T$.
\item[(ii)] For any subset $X \subseteq T$, $\bigvee X \in T$.
\item[(iii)] For any finite subset $X \subseteq T$, $\bigwedge X \in T$.
\end{itemize}
The elements of $T$ are called the open elements of $L$.
\end{defi}

Ok, without a complement operator, it will be hard to define what a closed element should be... Any comments? Weblinks for further reading?

[ reply | up ]

Re: topologies on complete lattices by yark on 2006-02-18 16:14:56

closed sets by vitriol on 2002-08-22 12:47:02

it might be interesting to note that you can equivalently define a topology in terms of it's closed sets, by demorgan's set laws. what do you think?

also, can you link the various examples of topological spaces to this entry? there's an examples thing now isn't there? i can't seem to find a definition for the topology induced by a metric space either, which is something you probably want to add.

Cheers,
Dave

[ reply | up ]

Re: closed sets by djao on 2002-08-22 23:51:46

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