## Time, Topology, Differential Logic • 5

And the founder, having shod a plough with a brazen ploughshare, and having yoked to it a bull and a cow, himself drove a deep furrow round the boundary lines, while those who followed after him had to turn the clods, which the plough threw up, inwards towards the city, and suffer no clod to lie turned outwards.

Plutarch • Life of Romulus

Re: Peirce List Discussion • ETETJBDJAJA

Our inquiry now calls on the rudiments of topology, for which I turn to J.L. Kelley.

### Chapter 1. Topological Spaces

#### 1.1. Topologies and Neighborhoods

A topology is a family $\mathcal{T}$ of sets which satisfies the two conditions:  the intersection of any two members of $\mathcal{T}$ is a member of $\mathcal{T},$ and the union of the members of each subfamily of $\mathcal{T}$ is a member of $\mathcal{T}.$  The set $X = \bigcup \{ U : U \in \mathcal{T} \}$ is necessarily a member of $\mathcal{T}$ because $\mathcal{T}$ is a subfamily of itself, and every member of $\mathcal{T}$ is a subset of ${X}.$  The set ${X}$ is called the space of the topology $\mathcal{T}$ and $\mathcal{T}$ is a topology for ${X}.$  The pair $(X, \mathcal{T})$ is a topological space.  When no confusion seems possible we may forget to mention the topology and write “${X}$ is a topological space.”  We shall be explicit in cases where precision is necessary (for example if we are considering two different topologies for the same set ${X}$).

The members of the topology $\mathcal{T}$ are called open relative to $\mathcal{T},$ or $\mathcal{T}$-open, or if only one topology is under consideration, simply open sets.  The space ${X}$ of the topology is always open, and the void set is always open because it is the union of the members of the void family.  These may be the only open sets, for the family whose only members are ${X}$ and the void set is a topology for ${X}.$  This is not a very interesting topology, but it occurs frequently enough to deserve a name;  it is called the indiscrete (or trivial) topology for ${X},$ and $(X, \mathcal{T})$ is then an indiscrete topological space.  At the other extreme is the family of all subsets of ${X},$ which is the discrete topology for ${X}$ (then $(X, \mathcal{T})$ is a discrete topological space).  If $\mathcal{T}$ is the discrete topology, then every subset of the space is open.  (Kelley, p. 37).

### References

• Kelley, J.L. (1955), General Topology, Van Nostrand Reinhold, New York, NY.
• Plutarch, “Romulus”, in Plutarch’s Lives : Volume 1, Bernadotte Perrin (trans.), Loeb Classical Library, William Heinemann, London, UK, 1914.

### 2 Responses to Time, Topology, Differential Logic • 5

1. I always enjoyed topology — we used Kelley when I took a course at DePaul as part of my MA. By the way, at some point you might want to look at Whitehead’s triptych of books on “natural philosophy” (always seemed a more appropriate term in Whitehead’s case than “philosophy of science”) from 1919 to 1922: “Enquiry into the Principles of Natural Knowledge,” “The Concept of Nature,” and “The Principle of Relativity.” The first two (“PNK” and “CN”, respectively) Whitehead introduces mereology into the analysis of space and time. (Lesniewski had brought mereology into play in 1916, but that was in Poland, and no one in Western Europe learned about it until many years later.) I discuss Whitehead’s approach in my book on the measurement problem of cosmology, but good luck laying your hands on that. Whitehead expands his mereology to a full blown mereotopology in “Process and Reality,” part IV, but that’s not a text for the faint of heart. A contemporary, and clearly formalized, characterization of Whitehead’s later theory was published by Bowman Clarke in a pair of essays in the Notre Dame Journal of Formal Logic, 1982 and 1984 if I recall.

I mention all of this because the topics of time, topology, and logic are quite central to Whitehead’s philosophy. Randy Auxier and I have a book coming out from Routledge, hopefully this Spring, and hopefully not to insanely expensive, that treats these developments of Whitehead’s in some depth.

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