# Dictionary Definition

pointlessness n : total lack of meaning or ideas
[syn: inanity, senselessness, mindlessness, vacuity]

# User Contributed Dictionary

## English

### Pronunciation

### Noun

pointlessness# Extensive Definition

In mathematics, pointless
topology (also called point-free or pointfree topology) is an
approach to topology
which avoids the mentioning of points.

## General concepts

Traditionally, a topological space consists of a set of points, together with a system of open sets. These open sets with the operations of intersection and union form a lattice with certain properties. Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points. Since some of the so-defined lattices do not arise from topological spaces, one may see the category of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces.## Categories of frames and locales

Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i.e. every (even infinite) subset of L has a supremum ⋁ai such that- b \wedge \left( \bigvee a_i\right) = \bigvee \left(a_i \wedge b\right)

for all b in L. These frames, together with
lattice homomorphisms which respect arbitrary suprema, form a
category. The dual
of the category of frames is called the category of locales and
generalizes the category
Top of all topological spaces with continuous functions. The
consideration of the dual category is motivated by the fact that
every
continuous map between topological spaces X and Y induces a map
between the lattices of open sets in the opposite direction as for
every continuous function
f: X → Y and every open set O
in Y the inverse
image f -1(O) is an open set in X''.

## Relation to point-set topology

It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems. While many important theorems in point-set topology require the axiom of choice, this is not true for their analogues in locale theory. This can be useful if one works in a topos which does not have the axiom of choice.The concept of "product of locales" diverges
slightly from the concept of "product of
topological spaces", and this divergence has been called a
disadvantage of the locale approach. Others claim that the locale
product is more natural and point to several of its "desirable"
properties which are not shared by products of topological
spaces.

For almost all spaces (more precisely for
sober
spaces) the topological product and the localic product have
the same set of points. The products differ in how equality between
sets of open rectangles (=the canonical base for the product
topology) is defined: equality for the topological product means
the same set of points is covered; equality for the localic product
means provable equality using the frame axioms. As a result two
open sublocales of a localic product may contain exactly the same
points without being equal.

A point where locale theory and topology diverge
much more strongly is the concept of subspaces vs. sublocales. The
rational numbers have c subspaces but 2c sublocales. The proof for
the latter statement is due to John Isbell and uses the fact that
the rational numbers have c many pairwise almost disjoint (= finite
intersection) closed subspaces.

## See also

- Heyting algebra. A locale is a complete Heyting algebra.
- Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between sober spaces and spatial locales, are to be found in the article on Stone duality.
- Point-free geometry
- Mereology
- Tacit programming

## References

- Johnstone, Peter T., 1983, "The point of pointless topology," Bulletin of the American Mathematical Society 8(1): 41-53.

pointlessness in Spanish: Topología sin
puntos

# Synonyms, Antonyms and Related Words

absurdity, aimlessness, aridity, barrenness, bloodlessness, bootlessness, characterlessness,
colorlessness,
deadness, dismalness, dragginess, dreariness, dryness, dullness, dustiness, effeteness, emptiness, etiolation, fatuity, fecklessness, flatness, fruitlessness, futility, heaviness, hollowness, impotence, inanity, ineffectiveness,
ineffectuality,
inefficacy, inexcitability, insipidity, insipidness, jejunity, leadenness, lifelessness, lowness of
spirit, meaninglessness,
nugacity, otiosity, paleness, pallor, pokiness, ponderousness, profitlessness, purposelessness, rat
race, slowness,
solemnity, spiritlessness, sterility, stiffness, stodginess, stuffiness, superficiality, tastelessness, tediousness, the absurd,
triviality, uninterestingness,
unliveliness,
unproductiveness,
unprofitability,
unprofitableness,
valuelessness,
vanity, vapidity, vapidness, vicious circle,
woodenness, worthlessness