What is an Entity? A Topological Definition

G. Stolyarov II
November 15, 2008
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Much of my work in A Rational Cosmology depends on a threefold distinction among existents.

Existents are either entities, qualities, or relationships, and each of these designations is mutually exclusive. I have been asked, however, what counts as an entity. In “Entities and Spatial Continuity,” I provided an explanation of one of the necessary qualities of an entity, which I called spatial continuity. Here is how I described that quality:

Every entity — homogeneous or heterogeneous — must have continuity among all of its parts. The test for spatial continuity is this: is it conceivable for one to trace a path from any point on the entity to any other point without any part of that path entering a region of “space-as-absence,” i.e., a region where the entity does not exist? If such a path is conceivable — no matter whether one’ s current level of technological advancement actually permits one to trace it — the entity is continuous and is affirmed in this ubiquitous quality.”

Having studied topology over the past several months, I was pleased to discover that a topological equivalent of my concept of spatial continuity exists. It is called path-connectedness.

Here is how James R. Munkres defines a path and path-connectedness in Chapter 24 of the Second Edition of his book, Topology:

Path: “Given points x and y of the space X, a path in X from x to y is a continuous map

f: [a, b] → X of some closed interval in the real line into X, such that f(a) = x and f(b) = y.”

Path Connectedness: “A space X is path connected if every pair of points of X can be joined by a path in X.”

A path-connected space is one in which any two points can be joined by a continuous function – a path – that never strays outside the space. To say that every entity is spatially continuous (as per my definition) is the same as saying that every entity is path-connected.

There is more that can be said about the ubiquitous qualities of entities by invoking the topological property of compactness. Compactness has a formal definition pertaining to every open cover of a space possessing a finite subcover, but for our purposes here, we need only to consider the Heine-Borel Theorem, which states that every closed and bounded subspace of an n-dimensional space of real numbers (Rn) is compact.

I argue in A Rational Cosmology that all entities are three-dimensional subspaces of R3 and that every entity has a finite nonzero volume and finite dimensions of length, width, and height, which means that every entity is bounded. Of course, every entity also includes its own boundary and so is closed in topological terms. Therefore, every entity is compact.

Therefore, we can use topology to concisely state the ubiquitous qualities of entities:

1) Every entity is a three-dimensional subspace of R3.

2) Every entity is path-connected.

3) Every entity is closed and bounded – and therefore compact.

4) Every entity exhibits the quality of matter (i.e., every entity is material). This is not a topological property, because topology only addresses spaces and not matter. However, Chapter XIV of A Rational Cosmology addresses my definition of matter in an accessible and concise fashion.

Any existent that meets the above four qualities is an entity; any presumed existent that does not is either not a genuine existent in itself or is a quality or relationship.

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Learn about Mr. Stolyarov's novel, Eden against the Colossus, here.

Read Mr. Stolyarov's comprehensive treatise, A Rational Cosmology, explicating such terms as the universe, matter, space, time, sound, light, life, consciousness, and volition, here.

Read Mr. Stolyarov's four-act play, Implied Consent, a futuristic intellectual drama on the sanctity of human life, here.