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Hometopological space
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topological space
A topological space is a set $X$ together with a set $\mathcal{T}$ whose elements are subsets of $X$, 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 $X$. The set $\mathcal{T}$ is called a topology on $X$. A subset $C\subset X$ is called a closed set if the complement $X\setminus C$ is an open set.
A topology $\mathcal{T}^{{\prime}}$ is said to be finer (respectively, coarser) than $\mathcal{T}$ if $\mathcal{T}^{{\prime}}\supset\mathcal{T}$ (respectively, $\mathcal{T}^{{\prime}}\subset\mathcal{T}$).
Examples

The discrete topology is the topology $\mathcal{T}=\mathcal{P}(X)$ on $X$, where $\mathcal{P}(X)$ denotes the power set of $X$. This is the largest, or finest, possible topology on $X$.

The indiscrete topology is the topology $\mathcal{T}=\{\emptyset,X\}$. It is the smallest or coarsest possible topology on $X$.
References
 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
 2 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
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Comments
closed sets
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
Re: closed sets
The examples are already parented to something else and multiparenting hasn't been implemented yet, so the most I can do is set them as related, which I have done.
Metric spaces have their own entry already. I will add a related link to the metric spaces entry.
As for closed sets, I suppose I will get around to it eventually. File an addendum to remind me about it.
topologies on complete lattices
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 nonreflexive 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?
Re: topologies on complete lattices
Schneemann writes:
> Can a topology be defined as a subset of an arbitrary complete
> (and complemented) lattice, instead of a power set?
There is a form of topology that deals only with a lattice of open sets, rather than with points. This is not quite the same as you're asking about, but you might want to take a look at it. Wikipedia has an article: http://en.wikipedia.org/wiki/Pointless_topology