# A simplified characterization of an open m-arrangement

## Full text

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ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X II (1969)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X II (1969)

## P roposition 1. Each flat is closed.

### P r o o f . Prom (ii), any (m — l)-fla t is closed. Any г-flat, 0 < i < m — 2, is the intersection of finitely many (m — l)-flats and hence is closed. Of course X and 0 are also closed.

1 — Prace matematyczne X II

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x

kj

w

w

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x

j

## P roposition 8. I f x, у and z are points of a 1-flat f, then \xy\ w \yz\

### = \®У\, \yz\, or \xz\.

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186 М. С. G e m i g n a n i

0

## Lemma 1. I ( Y )

=

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187

2

q

2

_ 2

### = F ° G (8) such that wex0u] given any u eE QY, x0u c Y since Y is convex.

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188 М. С. G r e m i g n a n i

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189

x

x

## Proposition11. a) F r Y = В ( Y ) = Bd(7(\$) (3.8).

### satisfy 3.1-3.9, hence X and G form an m-arrangement.

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190 М. С. G r e m i g n a n i

## Proposition12. I f 8 — {ж0, ..., xm} is a linearly independent subset of X , let A i be that component of X —f ( 8 — {xi}) which contains

Ш

0

m m

г=0

0

m

0

Ш

m г= о

г-0ш m

ъ=о г=о

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### Open m-arrangement

191

R e fe re n c e s

###  M. G e m ig n a n i,

Topological geometries and a new characterization of B m,

###  —

On eliminating an unwanted axiom from the characterisation of B m by means of topological geometries,

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