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More abstractly, a natural class of objects to study in topology are objects that are homogeneous ( all points are topologically the same: the group of self-homeomorphisms acts transitively ) and " finite type " or " tame " ( to rule out spaces such as the Cantor set, where each open set contains uncountably many connected components ); more generally, a space of " finite type " where the self-homeomorphism group has finitely many orbits, forming the strata.

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## Some Related Sentences

More and abstractly

__More__

__abstractly__

**,**Campbell's idea

**that**myths

**are**how we shape our lives deeply informs

**the**picture

**of**life

**in**Glorantha throughout

**the**game world's publication history

**.**

__More__

__abstractly__

**,**

**the**convex-hull operator Conv ()

**has**

**the**characteristic properties

**of**

**a**closure operator

**:**

__More__

__abstractly__

**,**one can define

**the**dicyclic

**group**Dic < sub > n </ sub >

**as**any

**group**having

**the**presentation

__More__

__abstractly__

**,**

**the**symplectic

**group**can be defined

**as**

**the**

**set**

**of**linear transformations

**of**

**a**2n-dimensional vector

**space**over F

**that**preserve

**a**nondegenerate

**,**skew-symmetric

**,**bilinear form

**.**

__More__

__abstractly__

**,**

**in**classical mechanics phase

**space**is

**the**cotangent

**space**

**of**configuration

**space**

**,**

**and**

**in**this interpretaton

**the**procedure above expresses

**that**

**a**choice

**of**local coordinates on configuration

**space**induces

**a**choice

**of**

**natural**local Darboux coordinates for

**the**standard symplectic structure on

**a**cotangent

**space**

**.**

__More__

__abstractly__

**and**

**more**precisely

**,**it may be taken

**to**ask whether specified axioms

**of**Euclidean geometry concerning

**the**existence

**of**lines

**and**circles entail

**the**existence

**of**

**such**

**a**square

**.**

__More__

__abstractly__

**,**

**the**integral can be written

**as**

**the**cap product

**of**

**the**homology

**class**

**of**Z

**and**

**the**cohomology

**class**represented by α

**.**

__More__

__abstractly__

**,**it can be defined

**as**an incidence structure

**(**P

**,**L

**,**I

**)**consisting

**of**

**a**

**set**P

**of**

**points**

**,**

**a**

**set**L

**of**lines

**,**

**and**an incidence relation I stating which

**points**lie on which lines

**.**

__More__

__abstractly__

**,**it means

**that**

**the**cohomology

**class**

**in**

**the**classifying

**space**for pulls back from

**the**cohomology

**class**

**in**under

**the**inclusion

**(**which corresponds

**to**

**the**inclusion

**and**similar ).

so too is

**the**dual**group**__More____abstractly__**,**these**are**both examples**of**representable functors**,**being represented respectively by K**and**T**.**
The order

**of**PSL**(**n**,**q**)**is**the**above**,**divided by**,****the**number**of**scalar matrices with determinant 1 –**or**equivalently dividing by**,****the**number**of**classes**of**element**that**have no nth root**,****or**equivalently**,**dividing by**the**number**of**nth roots**of**unity**in**.< ref**group**=" note "> These**are**equal because they**are****the**kernel**and**cokernel**of****the**endomorphism formally**,**__More____abstractly__**,****the**first realizes PSL**as**SL / SZ**,**while**the**second realizes PSL**as****the**kernel**of**__More__

**generally**

**,**if W is

**a**linear subspace

**of**

**a**

**(**possibly infinite dimensional

**)**vector

**space**V then

**the**codimension

**of**W

**in**V is

**the**dimension

**(**possibly infinite

**)**

**of**

**the**quotient

**space**V / W

**,**which is

**more**

__abstractly__known

**as**

**the**cokernel

**of**

**the**inclusion

**.**

__More__

__abstractly__

**,**it studies how global properties

**of**

**a**graph influence local substructures

**of**

**the**graph

**.**

__More__

__abstractly__

**,**

**the**mean curvature is

**the**trace

**of**

**the**second fundamental form divided by n

**(**

**or**equivalently

**,**

**the**shape operator ).

__More__

__abstractly__

**,**some aspects

**of**

**the**narrative appear

**in**

**the**crucifixion story

**of**Jesus found

**in**

**the**gospels

**.**

__More__

__abstractly__

**:**If X < sub > 1 </ sub >, ..., X < sub > n + 1 </ sub >

**are**conditionally independent random variables

**that**

**each**can assume

**the**value 0

**or**1

**,**then

**,**if we know nothing

**more**about them

**,**

__More__

__abstractly__

**and**

**generally**

**,**we note

**that**

**the**two quantities asserted

**to**be equal count

**the**subsets

**of**size k

**and**n − k

**,**respectively

**,**

**of**any n-element

**set**S

**.**There is

**a**simple bijection between

**the**two families F < sub > k </ sub >

**and**F < sub > n − k </ sub >

**of**subsets

**of**S

**:**it associates every k-element subset with its complement

**,**which

**contains**precisely

**the**remaining n − k elements

**of**S

**.**Since F < sub > k </ sub >

**and**F < sub > n − k </ sub > have

**the**

**same**number

**of**elements

**,**

**the**corresponding binomial coefficients must be equal

**.**

__More__

__abstractly__

**,**

**the**

**"**pay-off

**set**

**"**

**(**i

**.**e.,

**the**

**set**

**of**

**all**plays

**in**which

**the**angel wins

**)**is

**a**closed

**set**

**(**

**in**

**the**

**natural**

**topology**on

**the**

**set**

**of**

**all**plays ),

**and**it is known

**that**

**such**games

**are**determined

**.**

__More__

__abstractly__

**,**

**"**G-bundles over X

**"**is

**a**functor

**in**G

**:**given

**a**map H → G

**,**one gets

**a**map from H-bundles

**to**G-bundles by inducing

**(**

**as**above ).

More and natural

__More__recent observations made

**in**

**the**laboratory

**of**antibiosis between micro-organisms led

**to**

**the**discovery

**of**

__natural__antibacterials produced by microorganisms

**.**

__More__formally

**,**discrete mathematics

**has**been characterized

**as**

**the**branch

**of**mathematics dealing with countable sets

**(**sets

**that**have

**the**

**same**cardinality

**as**subsets

**of**

**the**

__natural__numbers

**,**including rational numbers but not real numbers ).

__More__idealized motion

**in**

**the**

**"**sublunary

**"**realm could only be achieved through artifice

**,**

**and**prior

**to**

**the**17th century

**,**

**many**did not view artificial experiments

**as**

**a**valid means

**of**learning about

**the**

__natural__world

**.**

__More__

**generally**

**,**

**a**meteorite on

**the**surface

**of**any celestial body is

**a**

__natural__object

**that**

**has**come from elsewhere

**in**

**space**

**.**

__More__substantive basis for this designation is

**that**

**the**vocabulary

**,**grammar

**,**

**and**orthography

**of**Interlingua

**are**

__natural__; they have been standardized

**and**presented by

**a**linguistic research body

**,**but they predated it

**and**

**are**not themselves considered

**a**product

**of**human invention

**.**

This can stem either from

__natural__causes**such****as****in**Olaf Stapledon's novel Odd John**,****and**Theodore Sturgeon's__More__Than Human**,****or**be**the**result**of**intentional augmentation**such****as****in**A**.**E**.**van Vogt's novel Slan**.**__More__precisely

**,**if A is

**a**

**finite**

**set**

**of**generators for G then

**the**word problem is

**the**membership problem for

**the**formal language

**of**

**all**words

**in**A

**and**

**a**formal

**set**

**of**inverses

**that**map

**to**

**the**identity under

**the**

__natural__map from

**the**free monoid with involution on A

**to**

**the**

**group**G

**.**If B is another

**finite**generating

**set**for G

**,**then

**the**word problem over

**the**generating

**set**B is equivalent

**to**

**the**word problem over

**the**generating

**set**A

**.**

__More__

**generally**

**,**

**the**term glacis can denote any slope

**,**

__natural__

**or**artificial

**,**which fulfils

**the**above requirements

**.**

__More__sophisticated devices which use

**a**graphical screen with touch sensing

**or**screen-edge buttons provide flexibility while minimizing

**space**used

**:**

**the**meaning

**of**

**the**buttons can change with

**the**screen

**,**

**and**selection involves

**the**

__natural__behavior

**of**pointing at what's desired

**.**

__More__specifically

**,**let be

**the**

**set**

**of**

**all**polynomials with integer coefficients

**(**

**such**

**a**polynomial is also referred

**to**

**as**

**a**polynomial over ), with addition

**and**multiplication defined

**to**be

__natural__polynomial addition

**and**multiplication

**.**

O ’ Neill notes

**that**Cavendish ’ s__natural__philosophy**,****and**writing**in**general**,**was criticized by**many****of**her contemporaries**as**well**as**by**more**recent readers**,****such****as**Samuel Pepys**,**Henry__More__**and**Virginia Woolf**.**__More__recently

**,**synthetic velvets have been developed

**,**mostly polyester

**,**nylon

**,**viscose

**,**acetate

**,**

**and**mixtures

**of**different synthetics

**,**

**or**synthetics

**and**

__natural__fibers

**(**for example viscose mixed with silk ).

__More__than half

**(**56 %)

**the**losses

**of**coastal wetlands resulted from dredging for marinas

**,**canals

**,**port development

**,**

**and**

**,**

**to**some extent

**,**from

__natural__shoreline erosion

**.**

__More__precisely

**,**it shows

**that**

**the**mere assumption

**of**

**the**existence

**of**

**the**

**set**

**of**

__natural__numbers

**as**

**a**totality

**(**an actual infinity

**)**suffices

**to**imply

**the**existence

**of**

**a**model

**(**

**a**world

**of**

**objects**

**)**

**of**any consistent theory

**.**

__More__than 1500 years had elapsed since

**the**Greek doctors

**,**under

**the**influence

**of**Hippocrates had examined illness

**as**

**a**

__natural__process

**.**

#

__More__**generally****,**an assignment**of**elements from**a**countable mathematical object**,****such****as****a**countable**group****,****to**__natural__numbers**to**allow algorithmic manipulation**of****the**mathematical object**.**__More__realistic

**,**

**and**often overlooked

**,**options include long-term crop rotations

**,**returning

**to**

__natural__cycles

**that**annually flood cultivated lands

**(**returning lost nutrients indefinitely

**)**

**such**

**as**

**the**Flooding

**of**

**the**Nile

**,**

**the**long-term use

**of**biochar

**,**

**and**use

**of**crop

**and**livestock landraces

**that**

**are**adapted

**to**less than ideal conditions

**such**

**as**pests

**,**drought

**,**

**or**lack

**of**nutrients

**.**

__More__than any other Spanish dramatist

**,**Alarcón was preoccupied with ethical aims

**,**

**and**his gift

**of**dramatic presentation is

**as**brilliant

**as**his dialogue is

__natural__

**and**vivacious

**.**

__More__than two-thirds

**of**

**that**growth can be ascribed

**to**

**a**

__natural__increase resulting from high fertility

**and**birthrates

**.**

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