[permalink] [id link]

Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf.

from
Wikipedia

## Some Related Sentences

Conversely and space

__Conversely__

**,**because

__space__

**is**expanding

**,**

**and**more distant objects are receding ever more quickly

**,**light emitted by us today may never " catch up " to very distant objects

**.**

* The spectrum of any bounded linear operator on

**a**Banach__space__**is****a**nonempty compact subset of**the**complex numbers C**.**__Conversely__**,**any compact subset of C arises in this manner**,**as**the**spectrum of some bounded linear operator**.**__Conversely__

**,**one might expect that inertial motions

**,**once identified by observing

**the**actual motions of bodies

**and**making allowances for

**the**external forces ( such as electromagnetism or friction ), can be used to define

**the**geometry of

__space__

**,**as well as

**a**time coordinate

**.**

__Conversely__

**,**in

**the**bleak Orwellian world of " The High Ones "— where

**the**Soviets have won

**the**Third World War

**and**gained control of

**the**whole world —

**the**dissidents still have some hope

**,**precisely because

__space__flight has not been abandoned

**.**

Given

**a**vector__space__V over**a**field K**,****the**span of**a**set S ( not necessarily finite )**is**defined to be**the**intersection W of all subspaces of V**which**contain S**.**W**is**referred to as**the**subspace spanned by S**,**or by**the**vectors in S**.**__Conversely__**,**S**is**called**a**spanning set of W**.**__Conversely__

**,**except in dimension 2

**,**

**if**

**the**curvature of

**a**Riemannian manifold has this form for some function K

**,**then

**the**Bianchi identities imply that K

**is**constant

**and**thus that

**the**manifold

**is**( locally )

**a**

__space__form

**.**

Equipped with this notion of length

**,****a**Riemannian manifold**is****a**metric__space__**,**meaning that**it**has**a**distance function d ( p**,**q ) whose value at**a**pair of points p**and**q**is****the**distance from p to q**.**__Conversely__**,****the**metric tensor itself**is****the**derivative of**the**distance function ( taken in**a**suitable manner ).__Conversely__

**,**every Polish

__space__

**is**homeomorphic to

**a**G < sub > δ </ sub >- subset of

**the**Hilbert cube

**.**

__Conversely__

**,**when

**a**record

**is**deleted

**,**later records are moved down so that

**the**free

__space__remains contiguous

**.**

__Conversely__

**,**

**if**X

**is**projective

**,**then

**the**pullback of under

**the**closed immersion of X into

**a**projective

__space__

**is**very ample

**.**

__Conversely__

**,**

**if**X

**is**

**a**Hausdorff

__space__

**and**ker f

**is**

**a**closed set

**,**then

**the**coimage of f

**,**

**if**given

**the**quotient

__space__topology

**,**must also be

**a**Hausdorff

__space__

**.**

Any point x gives rise to

**a**continuous function p < sub > x </ sub > from**the**one element topological__space__1 ( all subsets of**which**are open ) to**the**__space__X by defining p < sub > x </ sub >( 1 ) = x**.**__Conversely__**,**any function from 1 to X clearly determines one point:**the**element that**it**" points " to**.**__Conversely__

**,**any such one form defines ( via pullback )

**a**G-equivariant horizontal 1-form on P

**,**

**and**

**the**

__space__of principal G-connections

**is**an affine

__space__for this

__space__of 1-forms

**.**

Conversely and is

__Conversely__

**,**

**if**he gives

**a**heavy rating to his own reading

**,**

**and**finds more accurate facts in

**it**than in

**the**others

**,**

**a**point

__is__chalked up for

**the**intrinsic

**,**objective meaningfulness of this type of mediumistic material

**.**

__Conversely__

**,**an adiabatic process that

__is__irreversible

**and**extracts no work

__is__in an isenthalpic process

**,**such as viscous drag

**,**progressing towards

**a**nonnegative change in entropy

**.**

__Conversely__

**,**British English favours fitted as

**the**past tense of fit generally

**,**whereas

**the**preference of American English

__is__more complex: AmEng prefers fitted for

**the**metaphorical sense of having made an object " fit " ( i

**.**e., suited ) for

**a**purpose ; in spatial transitive contexts

**,**AmEng uses fitted for

**the**sense of having made an object conform to an unchanged object that

**it**surrounds ( e

**.**g., " fitted X around Y ") but fit for

**the**sense of having made an object conform to an unchanged object that surrounds

**it**( e

**.**g., " fit X into Y ");

**and**for

**the**spatial senses ( both intransitive

**and**transitive ) of having been matching with respect to contour

**,**with no alteration of either object implied

**,**AmEng prefers fit (" The clothes fit

**.**

__Conversely__

**the**use of true brass seems to have declined in Western Europe during this period in favour of gunmetals

**and**other mixed alloys but by

**the**end of

**the**first Millennium AD brass artefacts are found in Scandinavian graves in Scotland

**,**brass was being used in

**the**manufacture of coins in Northumbria

**and**there

__is__archaeological

**and**historical evidence for

**the**production of brass in Germany

**and**The Low Countries areas rich in calamine ore

**which**would remain important centres of brass making throughout

**the**medieval period

**,**especially Dinant – brass objects are still collectively known as dinanterie in French

**.**

__Conversely__

**,**

**if**

**a**Boolean ring A

__is__given

**,**we can turn

**it**into

**a**Boolean algebra by defining x ∨ y := x + y + ( x · y )

**and**x ∧ y := x · y

**.**

__Conversely__

**,**modern biological sciences ( including even concepts such as molecular ecology ) are intimately entwined

**and**dependent on

**the**methods developed through biotechnology

**and**what

__is__commonly thought of as

**the**life sciences industry

**.**

__Conversely__

**,**

**if**

**the**

**second**theory

__is__correct

**,**then

**the**Athenians were merely reacting to

**the**Persians attacking them

**.**

__Conversely__

**the**equilibrium position

__is__said to be " far to

**the**left "

**if**hardly any product

__is__formed from

**the**reactants

**.**

__Conversely__

**,**

**the**islands do not receive

**the**upwellings ( cold streams ) that affect

**the**West African coast

**,**so

**the**air temperature

__is__cooler than in Senegal

**,**but

**the**sea

__is__actually warmer

**,**because

**the**orographic relief of some islands

**,**as Sao Miguel with steep mountains

**,**cover

**it**with rich woods

**and**luxuriant vegetation where

**the**dense moisture condense

**and**soak

**the**plants

**,**rocks

**,**soil

**,**logs

**,**moss etc

**.**

__Conversely__in francophone Canada

**,**one hundredth of

**a**Canadian dollar

__is__informally called

**a**sou ( penny ), though cent

__is__official in both English

**and**French

**.**

__Conversely__

**,**an absence of secure cycle-parking

__is__

**a**recurring complaint by cyclists from cities with low modal share of cycling

**.**

Conversely and separable

__Conversely__

**,**

**if**

**is**

**a**

__separable__algebraic extension

**,**

**and**

**if**L

**is**any intermediate field

**,**then

**and**are

__separable__extensions

**.**

If f

**is**unramified at**a**point x**,**then f**is**quasi-finite at x**.**__Conversely__**,****if**f**is**quasi-finite at x**,****and****if**also**,****the**local ring of x in**the**fiber f < sup >− 1 </ sup >( f ( x )),**is****a**field**and****a**finite__separable__extension of κ ( f ( x )), then f**is**unramified at x .< ref > EGA IV < sub > 4 </ sub >, Théorème 17**.**4**.**1 .</ ref >

Conversely and if

__Conversely__

**,**

__if__statistics were uncovered

**which**contradicted

**a**cherished theory

**,**

**the**sources were denounced as faulty

**.**

__Conversely__

**,**experimenters would consider as impressive such statements as

**the**following

**,**

**which**

**,**

__if__they turned out to be hits

**,**are so unusual as to be really significant:

__Conversely__

**,**

**a**subset R defines

**a**binary function

__if__

**and**

**only**

__if__

**,**for any x in X

**and**y in Y

**,**there exists

**a**unique z in Z such that ( x

**,**y

**,**z ) belongs to R

**.**

__Conversely__

**,**

__if__ƒ: C → C

**is**

**a**function

**which**

**is**differentiable when regarded as

**a**function on R < sup > 2 </ sup >, then ƒ

**is**complex differentiable

__if__

**and**

**only**

__if__

**the**Cauchy – Riemann equations hold

**.**

__Conversely__

**,**

__if__

**the**amount of fat

**is**increased

**,**increasing

**the**number of calories ingested will

**only**add weight

**.**

__Conversely__

**,**

__if__

**the**source of waves

**is**moving away from

**the**observer

**,**each wave

**is**emitted from

**a**position farther from

**the**observer than

**the**previous wave

**,**so

**the**arrival time between successive waves

**is**increased

**,**reducing

**the**frequency

**.**

__Conversely__every partition of comes from an equivalence relation in this way

**,**according to

**which**

__if__

**and**

**only**

__if__

**and**belong to

**the**same set of

**the**partition

**.**

If neither A nor B includes

**the**idea of existence**,**then " some A are B " simply adjoins A to B**.**__Conversely__**,**__if__A or B do include**the**idea of existence in**the**way that " triangle " contains**the**idea " three angles equal to two right angles ", then " A exists "**is**automatically true**,****and**we have an ontological proof of A's existence**.**__Conversely__

**,**

__if__

**the**measured potential

**is**negative

**,**

**the**electrode

**is**more oxidizing than

**the**SHE ( such as Zn in ZnSO < sub > 4 </ sub > where

**the**standard electrode potential

**is**− 0

**.**76 V ).

__Conversely__

**,**

__if__

**the**firm

**is**able to get bulk discounts of an input

**,**then

**it**could have economies of scale in some range of output levels even

__if__

**it**has decreasing returns in production in that output range

**.**

__Conversely__

**,**daylight stock shot in daylight will look normal

**,**but orange

__if__shot under tungsten lights

**.**

__Conversely__

**,**

__if__

**the**data are sparse — that

**is**

**,**

__if__

**only**K out of N Fourier coefficients are nonzero — then

**the**complexity can be reduced to O ( K log N log ( N / K )),

**and**this has been demonstrated to lead to practical speedups compared to an ordinary FFT for N / K > 32 in

**a**large-N example ( N = 2 < sup > 22 </ sup >) using

**a**probabilistic approximate algorithm (

**which**estimates

**the**largest K coefficients to several decimal places ).

0.143 seconds.