Difference between revisions of "Topology"

Latest revision as of 09:28, 30 December 2016

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 $X$ is a collection of subsets, $J\subseteq\mathcal{P}(X)$^{[Note 1]} such that^[1][2]:

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

We call the elements of $\mathcal{J}$ the open sets of the topology.

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

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

Terminology

• For $x\in X$ we call $x$ a point (of the topological space $(X,\mathcal{ J })$)^[1]
• For $U\in\mathcal{J}$ we call $U$ an open set (of the topological space $(X,\mathcal{ J })$)^[1]

Examples

Given a set $X$, the following topologies can be constructed:

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

If $(X,d)$ 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 $(X,\preceq)$ is a poset, then we have the:

• Order topology

See also

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

Notes

1. Jump up ↑ Or $\mathcal{J}\in\mathcal{P}(\mathcal{P}(X))$ if you prefer, here $\mathcal{P}(X)$ denotes the power-set of $X$. This means that if $U\in\mathcal{J}$ then $U\subseteq X$

References

1. ↑ ^{Jump up to: 1.0 1.1 1.2} Introduction to Topological Manifolds - John M. Lee
2. Jump up ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha