Topology

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Caution:This page is about topologies only, usually when talk of topologies we don't mean a topology but rather a topological space which is a topology with its underlying set. See that page for more details

Definition

A topology on a set [ilmath]X[/ilmath] is a collection of subsets, [ilmath]J\subseteq\mathcal{P}(X)[/ilmath] such that:

• [ilmath]X\in\mathcal{J} [/ilmath] and [ilmath]\emptyset\in J[/ilmath]
• If [ilmath]\{U_i\}_{i=1}^n\subseteq\mathcal{J}[/ilmath] is a finite collection of elements of [ilmath]\mathcal{J} [/ilmath] then [ilmath]\bigcap_{i=1}^nU_i\in\mathcal{J}[/ilmath] too - [ilmath]\mathcal{J} [/ilmath] is closed under finite intersection.
• If [ilmath]\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J}[/ilmath] is any collection of elements of [ilmath]\mathcal{J} [/ilmath] (finite, countable, uncountable or otherwise) then [ilmath]\bigcup_{\alpha\in I}U_\alpha\in\mathcal{J}[/ilmath] - [ilmath]\mathcal{J} [/ilmath] is closed under arbitrary union.

We call the elements of [ilmath]\mathcal{J} [/ilmath] the open sets of the topology.

A topological space is simply a tuple consisting of a set (say [ilmath]X[/ilmath]) and a topology (say [ilmath]\mathcal{J} [/ilmath]) on that set - [ilmath](X,\mathcal{ J })[/ilmath].

Note: A topology may be defined in terms of closed sets - A closed set is a subset of [ilmath]X[/ilmath] whose complement is an open set. A subset of [ilmath]X[/ilmath] may be both closed and open, just one, or neither.

Terminology

• For [ilmath]x\in X[/ilmath] we call [ilmath]x[/ilmath] a point (of the topological space [ilmath](X,\mathcal{ J })[/ilmath])
• For [ilmath]U\in\mathcal{J} [/ilmath] we call [ilmath]U[/ilmath] an open set (of the topological space [ilmath](X,\mathcal{ J })[/ilmath])

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Examples

Given a set [ilmath]X[/ilmath], the following topologies can be constructed:

• Discrete topology - the topology here is [ilmath]\mathcal{P}(X)[/ilmath] - the power set of [ilmath]X[/ilmath].
• Indiscrete topology (AKA: Trivial topology) - the only open sets are [ilmath]X[/ilmath] itself and [ilmath]\emptyset[/ilmath]
• Finite complement topology - the open sets are [ilmath]\emptyset[/ilmath] and any set [ilmath]U\in\mathcal{P}(X)[/ilmath] such that [ilmath]\vert X-U\vert\in\mathbb{N} [/ilmath]

If [ilmath](X,d)[/ilmath] is a metric space, then we have the:

• Metric topology (AKA: topology induced by a metric) - whose open sets are exactly the ones we consider open in a metric sense
  • This uses open balls as a topological basis

If [ilmath](X,\preceq)[/ilmath] is a poset, then we have the:

• Order topology

See also

• Topological separation axioms
  • Covers things like Hausdorff space, Normal topological space, so forth.