Double origin topology

Not to be confused with Line with two origins.

In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane R² with an extra point, say 0*, added. In this case, the double origin topology gives a topology on the set X = R² ∐ {0*}, where ∐ denotes the disjoint union.

To give the set X a topology means to say which subsets of X are "open", and to do so in a way that the following axioms are met:^[1]

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. The set X and the empty set ∅ are open sets.

Construction

Given a point x belonging to X, such that x ≠ 0 and x ≠ 0*, the neighbourhoods of x are those given by the standard metric topology on R²−{0}.^[2] We define a countably infinite basis of neighbourhoods about the point 0 and about the additional point 0*. For the point 0, the basis, indexed by n, is defined to be:^[2]

$\ N(0,n) = \{ (x,y) \in {\bold R}^2 : x^2 + y^2 < 1/n^2, \ y > 0\} \cup \{0\} .$

In a similar way, the basis of neighbourhoods of 0* is defined to be:^[2]

$N(0^*,n) = \{ (x,y) \in {\bold R}^2 : x^2 + y^2 < 1/n^2, \ y < 0\} \cup \{0^*\} .$

Properties

The space R² ∐ {0*}, along with the double origin topology is an example of a Hausdorff space, although it is not completely Hausdorff. In terms of compactness, the space R² ∐ {0*}, along with the double origin topology fails to be either compact, paracompact or locally compact. Finally, it is an example of an arc connected space.^[3]

References

1. ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 3, ISBN 048668735X 
2. ^ ^{a b c} Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 92 − 93, ISBN 048668735X 
3. ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 198 – 199, ISBN 048668735X