Topology Words: Reasons for the particular names

[Cantor080]

From Munkres, Topology: "A topology on a set X is a collection T of subsets of X having the
following properties:
(1) ∅ and X are in T .
(2) The union of the elements of any subcollection of T is in T .
(3) The intersection of the elements of any finite subcollection of T is in T .
A set X for which a topology T has been specified is called a topological space

If X is a topological space with topology T , we say that a subset U of X is an
open set of X if U belongs to the collection T ."

I don't know why the elements of collection T are called as Open Sets here. Nothing seems to be open here to justify its name as there is justification of openness in open interval in real line, for the name to have word open.

[I am thinking to add and ask here, all the other topology related words, with the reason for their particular name (if there is any such reasoning).]

 

[jambaugh]

The openness of the set is the basis for all other definitions. What we need from topology is the definitions of convergence of sequences of points and continuity of mappings. Note that the Calculus I definition of continuity is that the value exists and that for every open neighborhood of the value (open interval of radius epsilon) there is a (small enough) open neighborhood (open interval of radius delta) of the point in the domain so that function maps every point in this domain set to the interior of the arbitrary open neighborhood of the value.

No matter how weird and wacky your topological space, you can define limits and continuity of mappings provided you know what the open sets are. The definition of "what are the open sets" defines the topology, it essentially IS the topology. Hence a topological space is a point set for which you know the open subsets. So "open set" then becomes an undefined term in the mathematics of topology along with "set" and "element" and the base generating notation. All the defined terms are defined in terms of these.

Sometimes but not always those open sets are defined via a metric.
The reason for going directly to the open sets is to be as general as possible, e.g. not relying on there existing a specific metric structure to tell you path lengths and thus distances between points.

 

[fresh_42]

Dedekind (1871) was the first who defined open sets. He described it with points and distances and called it Körper. The closest translation is probably body. As Dedekind and Cantor essentially built the dream team who started to develop both fields, set theory and topology, I assume that one of them created the term. At least their correspondence and publications should be the starting point of research, possibly Poincaré as well.

Btw., Dedekind's description might well be the reason to call it open, because: "if for any point there is a length such that all points closer than this belong to the system of points, too" settles a kind of open process. Closely related are the terms inner points, closed set, and limit points (innere Punkte, abgeschlossene Menge, Häufungspunkte) and I think one cannot answer the question for one of them without regarding the others. Topology (Listing, 1836 in a letter) had originally be named analysis situs (Leibniz). Hilbert already used the word topology in his famous discourse 1900 without feeling the necessity to explain it.

That's all I have found. So the answer to your question lies probably somewhere between 1870 and 1900, maybe 1920, and Dedekind seem's the right place to start searching.