nLab
bitopological space

Bitopological space

Bitopological space

Definitions

Recall that a topological space is a set XX equipped with a topological structure 𝒯\mathcal{T}. Well, a bitopological space is simply a set equipped with two topological structures (X,𝒯,𝒯 *)(X, \mathcal{T}, \mathcal{T}^*). Unlike with bialgebras, no compatibility condition is required between these structures.

A bicontinous map is a function between bitopological spaces that is continuous with respect to each topological structure.

Bitopological spaces and bicontinuous maps form a category BiTopBiTop.

Separation axioms for topologies

Let ClCl denote the closure operator with respect to 𝒯\mathcal{T} and let Cl *Cl^* denote the closure operator with respect to 𝒯 *\mathcal{T}^*.

Proposition

Let (X,𝒯,𝒯 *)(X, \mathcal{T}, \mathcal{T}^*) be a bitopological space. Consider the following properties of this space:

  1. for each point xx there is a 𝒯\mathcal{T}-neighborhood base consisting of 𝒯 *\mathcal{T}^*-closed sets;

  2. for all xXx\in X and all 𝒯\mathcal{T}-opens UU containing xx there is a 𝒯 *\mathcal{T}^*-closed 𝒯\mathcal{T}-neighborhood VV of xx such that VU V \subset U;

  3. Cl *(O)Cl(O)Cl^*(O) \subset Cl(O) for each 𝒯 *\mathcal{T}^*-open OO;

  4. for all xXx\in X and all 𝒯\mathcal{T}-neighborhoods UU of xx the closure Cl *(U)Cl^*(U) is a 𝒯 *\mathcal{T}^*-neighborhood;

  5. for each point xx and each 𝒯\mathcal{T}-closed 𝒯\mathcal{T}-neighborhood VV of xx in XX there exists a 𝒯 *\mathcal{T}^*-closed 𝒯\mathcal{T}-neighborhood UU of xx in XX such that UU is contained in VV.

There are the following implications among these properties

Especially, all properties are equivalent if 𝒯\mathcal{T} is regular.

Proof

(1) \iff (2): Given a neighborhood base for a point xx as guaranteed by the first property. When you spell out the properties of this neighborhood base, you end up with the second property. For the reverse direction start with an arbitrary 𝒯\mathcal{T}-neighborhood base of a point xx consisting of open. Apply the second property to every element of this neighborhood base to the desired neighborhood base.

(3) \iff (4): Suppose property (3), and let UU be a 𝒯\mathcal{T}-neighborhood of an arbitrary point xx. Then the complement Cl *(U)˜ \widetilde{Cl^*(U)} is in 𝒯 *\mathcal{T}^*, so that Cl *(Cl *U˜)Cl(Cl *U˜) Cl^*(\widetilde{Cl^*U}) \subset Cl(\widetilde{Cl^* U}) by the first property. Hence Cl(Cl *U˜)˜Cl *(Cl *U˜)˜ \widetilde{Cl(\widetilde{Cl^*U})} \subset \widetilde{Cl^*(\widetilde{Cl^* U})} for the complements. Since UU is a 𝒯\mathcal{T}-neighborhood of xx, xx does not belong to Cl(Cl *U˜)Cl(\widetilde{Cl^*U}). Moreover, Cl *(Cl *U˜)˜\widetilde{Cl^*(\widetilde{Cl^*U})} is 𝒯 *\mathcal{T}^*-open and a subset of Cl *(U) Cl^*(U). Hence Cl *(U)Cl^*(U) is a 𝒯 *\mathcal{T}^*-neighborhood of xx.

For the converse suppose property (4). Let OO be a nonempty 𝒯 *\mathcal{T}^*-open set and xx an element of Cl *(O)Cl^*(O). Then if UU is any 𝒯\mathcal{T}-neighborhood of xx, some point yOy \in O belongs to Cl *(U)Cl^*(U) due to the second property. Hence, as OO is a 𝒯 *\mathcal{T}^*-neighborhood of yy, some point of UU belongs to OO. Thus xCl(O)x \in Cl(O), and therefore Cl *(G)Cl(O)Cl^*(G) \subset Cl(O).

(1) \implies (4): Given xXx\in X and a 𝒯\mathcal{T}-neighborhood UU by property (1) there is a 𝒯 *\mathcal{T}^*-open OUO \subset U containing xx. Hence OCl *(U)O \subset Cl^*(U), and Cl *(U)Cl^*(U) is a 𝒯 *\mathcal{T}^*-neighborhood.

(3) and 𝒯\mathcal{T} regular \implies (2): Let xXx\in X and UU be a 𝒯\mathcal{T}-open containing xx. By regularity of 𝒯\mathcal{T} we can find disjoint 𝒯\mathcal{T}-opens VxV' \ni x and UU˜U' \supset \tilde{U} (U˜\tilde{U} denotes the complement). Set VCl *(V)V \coloneqq Cl^*(V'). This set is obviously a 𝒯 *\mathcal{T}^*-closed 𝒯\mathcal{T}-neighborhood of xx. Due to property (3) VCl(V)V \subset Cl(V'). Since also Cl(V)U˜Cl(V') \subset \widetilde{U'}, we have VUV \subset U. This is to say that VV is the 𝒯\mathcal{T}-neighborhood we sought.

(5) and 𝒯\mathcal{T} regular \implies (2): Let xXx\in X and UU be a 𝒯\mathcal{T}-open containing xx. By regularity of 𝒯\mathcal{T} we can find disjoint 𝒯\mathcal{T}-opens VxV' \ni x and UU˜U' \supset \tilde{U} (U˜\tilde{U} denotes the complement). Due to property (5) the closed set U˜\widetilde{U'} contains a 𝒯 *\mathcal{T}^*-closed neighborhood of xx. This is the neighborhood we sought.

(1) \implies (5): Given some 𝒯\mathcal{T}-closed 𝒯\mathcal{T}-neighborhood VV of some point xx choose a neighborhood base according to property (1) and take an element UU therein that is contained in VV.

Definition

Let (X,𝒯,𝒯 *)(X, \mathcal{T}, \mathcal{T}^*) be a bitopological space. The topology 𝒯\mathcal{T} is regular with respect to 𝒯 *\mathcal{T}^* if one of the two equivalent conditions (1) and (2) from proposition holds. A bitopological space (X,𝒯,𝒯 *)(X, \mathcal{T}, \mathcal{T}^*) is called pairwise regular if 𝒯\mathcal{T} is regular with respect to 𝒯 *\mathcal{T}^* and vise versa.

Definition

Let (X,𝒯,𝒯 *)(X, \mathcal{T}, \mathcal{T}^*) be a bitopological space. The topology 𝒯 *\mathcal{T}^* is coupled to 𝒯\mathcal{T} if one of the two equivalent conditions (3) and (4) from proposition holds.

Not that is if 𝒯 *\mathcal{T}^* is coupled to a finer topology 𝒯𝒯 *\mathcal{T} \supset \mathcal{T}^* then 𝒯 *\mathcal{T}^* is coupled to every topology coarser than 𝒯\mathcal{T} due to property (3). Moreover in this case also 𝒯\mathcal{T} is coupled to 𝒯 *\mathcal{T}^* (again a direct consequence of property (3)).

Definition

Let (X,𝒯,𝒯 *)(X, \mathcal{T}, \mathcal{T}^*) be a bitopological space. The topology 𝒯 *\mathcal{T}^* is called a cotopology of 𝒯\mathcal{T} if 𝒯 *𝒯\mathcal{T}^* \subseteq \mathcal{T} and property (5) from proposition holds. The space (X,𝒯 *)(X, \mathcal{T}^*) is also called a cospace of (X,𝒯(X, \mathcal{T}.

Remarks

It is interesting and perhaps surprising that many advanced topological notions can be described using bitopological spaces, even when you would not naïvely think that there are two topologies around. (At least, that's my vague memory of what they were good for. I think that this was in some article by Isbell.)

References

  • Wikipedia entry

  • Jiri Adamek, Horst Herrlich, and George Strecker, Abstract and Concrete Categories: The Joy of Cats, Dover New York 2009. (pdf) pp. 59-60, 278

  • B. Dvalishvili, Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures and Applications, Elsevier Amsterdam 2005.

  • Peter Johnstone, Collapsed Toposes as Bitopological Spaces, pp. 19-35 in Categorical Topology, World Scientific Singapore 1989.

  • O. K. Klinke, A. Jung, A. Moshier, A bitopological point-free approach to compactications (2011). (preprint)

  • R. Kopperman, Asymmetry and duality in topology, Topology Appl. 66 no. 1 (1995) pp. 1-39.

The idea naturally appeared first in the context of quasi-metric spaces

  • Wallace Alvin Wilson, On Quasi-Metric Spaces, American Journal of Mathematics (1931), vol. 53, no. 3, pp. 675-684.

The notions of separation axioms were introduced in

  • J. D. Weston, On the comparison of topologies 1956, Journal of the London Mathematical Society, vol. s1-32 no. 3, pp. 342-354,

  • J. C. Kelly, Bitopological spaces, Proc. London Math. Soc. 13 no.3 (1963) pp. 71-89.

Only Kelly introduced the concept in its nowadays formulation of a set equipped with two topologies. The Russian school contributed the following comprehensive overviews of this and related topics

  • A. A. Ivanov, Problems of the Theory of Bitopological Spaces, 1990, Journal of Soviet Mathematics, vol. 52, Issue 1, pp. 2759-2790. Originally published as Проблематика теории битопологических пространств in Zap. Nauchn. Sem. POMI, 1988, vol. 167 (Russian version).

  • A. A. Ivanov, Problems of the Theory of Bitopological Spaces 2, 1996, Journal of Math. Sciences, vol. 81, Issue 2. Originally publishes as Проблематика теории битопологических пространств. 2 in Zap. Nauchn. Sem. POMI, 1993, Volume 208, pp. 5–67 (Russian version).

  • A. A. Ivanov, Problems of the Theory of Bitopological Spaces 3, 1998, Journal of Math. Sciences, vol. 91, Issue 6, pp 3339–3364. Originally published as Проблематика теории битопологических пространств. 3 in Zap. Nauchn. Sem. POMI, 1995, Volume 231, pp. 9–54 (Russian version).

as well as a more introductory text book

  • Битопологические пространства, 1997. Исследования по топологии. 9, Zap. Nauchn. Sem. POMI, 242, editor A. A. Ivanov (Russian version).

Last revised on March 31, 2019 at 04:40:46. See the history of this page for a list of all contributions to it.