In
mathematics, in the field of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is said to be a door space if every subset is
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
or
closed (or
both
Both may refer to:
Common English word
* ''both'', a determiner or indefinite pronoun denoting two of something
* ''both... and'', a correlative conjunction
People
* Both (surname)
Music
* The Both, an American musical duo consisting of ...
).
[Kelley, ch.2, Exercise C, p. 76.] The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither".
Properties
Here are some facts about door spaces:
* A
Hausdorff door space has at most one
accumulation point.
* In a
Hausdorff door space if
is not an
accumulation point then
is open.
To prove the first assertion, let
be a Hausdorff door space, and let
be distinct points. Since
is Hausdorff there are open neighborhoods
and
of
and
respectively such that
Suppose
is an accumulation point. Then
is closed, since if it were open, then we could say that
is open, contradicting that
is an accumulation point. So we conclude that as
is closed,
is open and hence