Metrizable topological vector space explained
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
Pseudometrics and metrics
A pseudometric on a set
is a map
satisfying the following properties:
-
;
- Symmetry:
d(x,y)=d(y,x)forallx,y\inX
; - Subadditivity:
d(x,z)\leqd(x,y)+d(y,z)forallx,y,z\inX.
A pseudometric is called a metric if it satisfies:
- Identity of indiscernibles: for all
if
then
Ultrapseudometric
A pseudometric
on
is called a
ultrapseudometric or a
strong pseudometric if it satisfies:
- Strong/Ultrametric triangle inequality:
d(x,z)\leqmax\{d(x,y),d(y,z)\}forallx,y,z\inX.
Pseudometric space
A pseudometric space is a pair
consisting of a set
and a pseudometric
on
such that
's topology is identical to the topology on
induced by
We call a pseudometric space
a
metric space (resp.
ultrapseudometric space) when
is a metric (resp. ultrapseudometric).
Topology induced by a pseudometric
If
is a pseudometric on a set
then collection of
open balls:
as
ranges over
and
ranges over the positive real numbers,forms a basis for a topology on
that is called the
-topology or the
pseudometric topology on
induced by
If
is a pseudometric space and
is treated as a
topological space, then unless indicated otherwise, it should be assumed that
is endowed with the topology induced by
Pseudometrizable space
A topological space
is called
pseudometrizable (resp.
metrizable,
ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric)
on
such that
is equal to the topology induced by
Pseudometrics and values on topological groups
An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.
A topology
on a real or complex vector space
is called a
vector topology or a
TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes
into a
topological vector space).
Every topological vector space (TVS)
is an additive commutative topological group but not all group topologies on
are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space
may fail to make scalar multiplication continuous. For instance, the
discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.
Translation invariant pseudometrics
If
is an additive group then we say that a pseudometric
on
is
translation invariant or just
invariant if it satisfies any of the following equivalent conditions:
- Translation invariance:
d(x+z,y+z)=d(x,y)forallx,y,z\inX
; d(x,y)=d(x-y,0)forallx,y\inX.
Value/G-seminorm
If
is a
topological group the a
value or
G-seminorm on
(the
G stands for Group) is a real-valued map
with the following properties:
- Non-negative:
- Subadditive:
p(x+y)\leqp(x)+p(y)forallx,y\inX
; -
- Symmetric:
where we call a G-seminorm a G-norm if it satisfies the additional condition:
- Total/Positive definite: If
then
Properties of values
If
is a value on a vector space
then:
|p(x)-p(y)|\leqp(x-y)forallx,y\inX.
-
and
for all
and positive integers
- The set
is an additive subgroup of
An invariant pseudometric that doesn't induce a vector topology
Let
be a non-trivial (i.e.
) real or complex vector space and let
be the translation-invariant
trivial metric on
defined by
and
such that
The topology
that
induces on
is the
discrete topology, which makes
into a commutative topological group under addition but does form a vector topology on
because
is
disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on
This example shows that a translation-invariant (pseudo)metric is enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms.
Additive sequences
A collection
of subsets of a vector space is called
additive if for every
there exists some
such that
All of the above conditions are consequently a necessary for a topology to form a vector topology. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups.
Assume that
n\bull=\left(n1,\ldots,nk\right)
always denotes a finite sequence of non-negative integers and use the notation:
For any integers
and
From this it follows that if
n\bull=\left(n1,\ldots,nk\right)
consists of distinct positive integers then
\sum
\subseteq
U | |
| -1+min\left(n\bull\right) |
.
It will now be shown by induction on
that if
n\bull=\left(n1,\ldots,nk\right)
consists of non-negative integers such that
for some integer
then
This is clearly true for
and
so assume that
which implies that all
are positive. If all
are distinct then this step is done, and otherwise pick distinct indices
such that
and construct
m\bull=\left(m1,\ldots,mk-1\right)
from
by replacing each
with
and deleting the
element of
(all other elements of
are transferred to
unchanged). Observe that
and
(because
) so by appealing to the inductive hypothesis we conclude that
\sum
\subseteq\sum
\subseteqUM,
as desired.
It is clear that
and that
so to prove that
is subadditive, it suffices to prove that
when
are such that
which implies that
This is an exercise. If all
are symmetric then
if and only if
from which it follows that
and
If all
are balanced then the inequality
for all unit scalars
such that
is proved similarly. Because
is a nonnegative subadditive function satisfying
as described in the article on
sublinear functionals,
is uniformly continuous on
if and only if
is continuous at the origin. If all
are neighborhoods of the origin then for any real
pick an integer
such that
so that
implies
If the set of all
form basis of balanced neighborhoods of the origin then it may be shown that for any
there exists some
such that
implies
Paranorms
If
is a vector space over the real or complex numbers then a
paranorm on
is a G-seminorm (defined above)
on
that satisfies any of the following additional conditions, each of which begins with "for all sequences
in
and all convergent sequences of scalars
":
- Continuity of multiplication: if
is a scalar and
are such that
and
then
p\left(sixi-sx\right)\to0.
- Both of the conditions:
and if
is such that
then
;
then
for every scalar
- Both of the conditions:
and
for some scalar
then
;
then
p\left(six\right)\to0forallx\inX.
- Separate continuity:
for some scalar
then
for every
;
is a scalar,
and
then
.
A paranorm is called total if in addition it satisfies:
- Total/Positive definite:
implies
Properties of paranorms
If
is a paranorm on a vector space
then the map
defined by
is a translation-invariant pseudometric on
that defines a on
If
is a paranorm on a vector space
then:
- the set
is a vector subspace of
-
with
- If a paranorm
satisfies
p(sx)\leq|s|p(x)forallx\inX
and scalars
then
is absolutely homogeneity (i.e. equality holds) and thus
is a seminorm.
Examples of paranorms
- If
is a translation-invariant pseudometric on a vector space
that induces a vector topology
on
(i.e.
is a TVS) then the map
defines a continuous paranorm on
; moreover, the topology that this paranorm
defines in
is
- If
is a paranorm on
then so is the map
- Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
- Every seminorm is a paranorm.
- The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).
- The sum of two paranorms is a paranorm.
- If
and
are paranorms on
then so is
(p\wedgeq)(x):=inf\{p(y)+q(z):x=y+zwithy,z\inX\}.
Moreover,
and
This makes the set of paranorms on
into a conditionally complete lattice. - Each of the following real-valued maps are paranorms on
:
- The real-valued maps
(x,y)\mapsto\sqrt{\left|x2-y2\right|}
and (x,y)\mapsto\left|x2-y2\right|3/2
are a paranorms on
- If
is a Hamel basis on a vector space
then the real-valued map that sends
(where all but finitely many of the scalars
are 0) to
\sumi\sqrt{\left|si\right|}
is a paranorm on
which satisfies
for all
and scalars
- The function
p(x):=|\sin(\pix)|+min\{2,|x|\}
is a paranorm on
that is balanced but nevertheless equivalent to the usual norm on
Note that the function
is subadditive. - Let
be a complex vector space and let
denote
considered as a vector space over
Any paranorm on
is also a paranorm on
F-seminorms
If
is a vector space over the real or complex numbers then an
F-seminorm on
(the
stands for
Fréchet) is a real-valued map
with the following four properties:
- Non-negative:
- Subadditive:
for all
- Balanced:
for
all scalars
satisfying
- This condition guarantees that each set of the form
or
for some
is a balanced set. - For every
p\left(\tfrac{1}{n}x\right)\to0
as
| infty |
\left(\tfrac{1}{n}\right) | |
| n=1 |
can be replaced by any positive sequence converging to the zero.
An F-seminorm is called an F-norm if in addition it satisfies:
- Total/Positive definite:
implies
An F-seminorm is called monotone if it satisfies:
- Monotone:
for all non-zero
and all real
and
such that
F-seminormed spaces
An F-seminormed space (resp. F-normed space) is a pair
consisting of a vector space
and an
F-seminorm (resp.
F-norm)
on
If
and
are
F-seminormed spaces then a map
is called an
isometric embedding if
q(f(x)-f(y))=p(x,y)forallx,y\inX.
Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.
Examples of F-seminorms
- Every positive scalar multiple of an F-seminorm (resp. F-norm, seminorm) is again an F-seminorm (resp. F-norm, seminorm).
- The sum of finitely many F-seminorms (resp. F-norms) is an F-seminorm (resp. F-norm).
- If
and
are F-seminorms on
then so is their pointwise supremum
x\mapsto\sup\{p(x),q(x)\}.
The same is true of the supremum of any non-empty finite family of F-seminorms on
- The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).
- A non-negative real-valued function on
is a seminorm if and only if it is a convex F-seminorm, or equivalently, if and only if it is a convex balanced G-seminorm. In particular, every seminorm is an F-seminorm.
- For any
the map
on
defined by is an F-norm that is not a norm.
- If
is a linear map and if
is an F-seminorm on
then
is an F-seminorm on
- Let
be a complex vector space and let
denote
considered as a vector space over
Any F-seminorm on
is also an F-seminorm on
Properties of F-seminorms
Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm. Every F-seminorm on a vector space
is a value on
In particular,
and
for all
Topology induced by a family of F-seminorms
Suppose that
is a non-empty collection of
F-seminorms on a vector space
and for any finite subset
and any
let
The set
\left\{Ul{F,r}~:~r>0,l{F}\subseteql{L},l{F}finite\right\}
forms a filter base on
that also forms a neighborhood basis at the origin for a vector topology on
denoted by
}. Each
is a
balanced and
absorbing subset of
These sets satisfy
-
} is the coarsest vector topology on
making each
continuous.
-
} is Hausdorff if and only if for every non-zero
there exists some
such that
- If
is the set of all continuous F-seminorms on
}\right) then
} = \tau_.
- If
is the set of all pointwise suprema of non-empty finite subsets of
of
then
is a directed family of F-seminorms and
} = \tau_.
Fréchet combination
Suppose that
is a family of non-negative subadditive functions on a vector space
The Fréchet combination of
is defined to be the real-valued map
As an F-seminorm
Assume that
is an increasing sequence of seminorms on
and let
be the Fréchet combination of
Then
is an
F-seminorm on
that induces the same locally convex topology as the family
of seminorms.
Since
is increasing, a basis of open neighborhoods of the origin consists of all sets of the form
\left\{x\inX~:~pi(x)<r\right\}
as
ranges over all positive integers and
ranges over all positive real numbers.
The translation invariant pseudometric on
induced by this
F-seminorm
is
This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.
As a paranorm
If each
is a paranorm then so is
and moreover,
induces the same topology on
as the family
of paranorms. This is also true of the following paranorms on
:
q(x):=inf\left\{
pi(x)+
~:~n>0isaninteger\right\}.
r(x):=
min\left\{
,pn(x)\right\}.
Generalization
The Fréchet combination can be generalized by use of a bounded remetrization function.
A is a continuous non-negative non-decreasing map
that has a bounded range, is
subadditive (meaning that
for all
), and satisfies
if and only if
Examples of bounded remetrization functions include
and
If
is a pseudometric (respectively, metric) on
and
is a bounded remetrization function then
is a bounded pseudometric (respectively, bounded metric) on
that is uniformly equivalent to
Suppose that
is a family of non-negative
F-seminorm on a vector space
is a bounded remetrization function, and
is a sequence of positive real numbers whose sum is finite. Then
defines a bounded
F-seminorm that is uniformly equivalent to the
It has the property that for any net
in
if and only if
pi\left(x\bull\right)\to0
for all
is an
F-norm if and only if the
separate points on
Characterizations
Of (pseudo)metrics induced by (semi)norms
A pseudometric (resp. metric)
is induced by a seminorm (resp. norm) on a vector space
if and only if
is translation invariant and
absolutely homogeneous, which means that for all scalars
and all
in which case the function defined by
is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by
is equal to
Of pseudometrizable TVS
If
is a
topological vector space (TVS) (where note in particular that
is assumed to be a vector topology) then the following are equivalent:
-
is pseudometrizable (i.e. the vector topology
is induced by a pseudometric on
).
-
has a countable neighborhood base at the origin.
- The topology on
is induced by a translation-invariant pseudometric on
- The topology on
is induced by an F-seminorm.
- The topology on
is induced by a paranorm.
Of metrizable TVS
If
is a TVS then the following are equivalent:
-
is metrizable.
-
is Hausdorff and pseudometrizable.
-
is Hausdorff and has a countable neighborhood base at the origin.
- The topology on
is induced by a translation-invariant metric on
- The topology on
is induced by an F-norm.
- The topology on
is induced by a monotone F-norm.
- The topology on
is induced by a total paranorm.
Of locally convex pseudometrizable TVS
If
is TVS then the following are equivalent:
-
is locally convex and pseudometrizable.
-
has a countable neighborhood base at the origin consisting of convex sets.
- The topology of
is induced by a countable family of (continuous) seminorms.
- The topology of
is induced by a countable increasing sequence of (continuous) seminorms
(increasing means that for all
- The topology of
is induced by an F-seminorm of the form: where
are (continuous) seminorms on
Quotients
Let
be a vector subspace of a topological vector space
- If
is a pseudometrizable TVS then so is
- If
is a complete pseudometrizable TVS and
is a closed vector subspace of
then
is complete.
- If
is metrizable TVS and
is a closed vector subspace of
then
is metrizable.
- If
is an F-seminorm on
then the map
defined byis an F-seminorm on
that induces the usual quotient topology on
If in addition
is an F-norm on
and if
is a closed vector subspace of
then
is an F-norm on
Examples and sufficient conditions
is pseudometrizable with a canonical pseudometric given by
for all
.
If
is pseudometric TVS with a translation invariant pseudometric
then
defines a paranorm. However, if
is a translation invariant pseudometric on the vector space
(without the addition condition that
is), then
need not be either an F-seminorm nor a paranorm.If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.Suppose
is either a DF-space or an LM-space. If
is a sequential space then it is either metrizable or else a Montel DF-space.If
is Hausdorff locally convex TVS then
with the
strong topology,
\left(X,b\left(X,X\prime\right)\right),
is metrizable if and only if there exists a countable set
of bounded subsets of
such that every bounded subset of
is contained in some element of
of a metrizable locally convex space (such as a
Fréchet space[1])
is a
DF-space. The strong dual of a DF-space is a
Fréchet space. The strong dual of a
reflexive Fréchet space is a
bornological space. The strong bidual (that is, the
strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.If
is a metrizable locally convex space then its strong dual
has one of the following properties, if and only if it has all of these properties: (1)
bornological, (2)
infrabarreled, (3)
barreled.
Normability
A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin. Moreover, a TVS is normable if and only if it is Hausdorff and seminormable. Every metrizable TVS on a finite-dimensional vector space is a normable locally convex complete TVS, being TVS-isomorphic to Euclidean space. Consequently, any metrizable TVS that is normable must be infinite dimensional.
If
is a metrizable
locally convex TVS that possess a
countable fundamental system of bounded sets, then
is normable.
If
is a Hausdorff
locally convex space then the following are equivalent:
-
is normable.
-
has a (von Neumann) bounded neighborhood of the origin.
- the strong dual space
of
is normable.
and if this locally convex space
is also metrizable, then the following may be appended to this list:
- the strong dual space of
is metrizable.
- the strong dual space of
is a Fréchet–Urysohn locally convex space.[1]
In particular, if a metrizable locally convex space
(such as a
Fréchet space) is normable then its
strong dual space
is not a
Fréchet–Urysohn space and consequently, this
complete Hausdorff locally convex space
is also neither metrizable nor normable.
Another consequence of this is that if
is a
reflexive locally convex TVS whose strong dual
is metrizable then
is necessarily a reflexive Fréchet space,
is a
DF-space, both
and
are necessarily
complete Hausdorff
ultrabornological distinguished webbed spaces, and moreover,
is normable if and only if
is normable if and only if
is Fréchet–Urysohn if and only if
is metrizable. In particular, such a space
is either a
Banach space or else it is not even a Fréchet–Urysohn space.
Metrically bounded sets and bounded sets
Suppose that
is a pseudometric space and
The set
is
metrically bounded or
-bounded if there exists a real number
such that
for all
; the smallest such
is then called the
diameter or
-diameter of
If
is
bounded in a pseudometrizable TVS
then it is metrically bounded; the converse is in general false but it is true for
locally convex metrizable TVSs.
Properties of pseudometrizable TVS
- Every metrizable locally convex TVS is a quasibarrelled space, bornological space, and a Mackey space.
- Every complete metrizable TVS is a barrelled space and a Baire space (and hence non-meager). However, there exist metrizable Baire spaces that are not complete.
- If
is a metrizable locally convex space, then the strong dual of
is bornological if and only if it is barreled, if and only if it is infrabarreled.
- If
is a complete pseudometrizable TVS and
is a closed vector subspace of
then
is complete.
- The strong dual of a locally convex metrizable TVS is a webbed space.
- If
and
are complete metrizable TVSs (i.e. F-spaces) and if
is coarser than
then
; this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete. Said differently, if
and
are both F-spaces but with different topologies, then neither one of
and
contains the other as a subset. One particular consequence of this is, for example, that if
is a Banach space and
is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of
(i.e. if
or if
for some constant
), then the only way that
can be a Banach space (i.e. also be complete) is if these two norms
and
are equivalent; if they are not equivalent, then
can not be a Banach space. As another consequence, if
is a Banach space and
is a Fréchet space, then the map
is continuous if and only if the Fréchet space
the TVS
(here, the Banach space
is being considered as a TVS, which means that its norm is "forgetten" but its topology is remembered).
- A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space.[1]
- Any product of complete metrizable TVSs is a Baire space.
- A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension
- A product of pseudometrizable TVSs is pseudometrizable if and only if it all but at most countably many of these TVSs have the trivial topology.
- Every complete metrizable TVS is a barrelled space and a Baire space (and thus non-meager).
- The dimension of a complete metrizable TVS is either finite or uncountable.
Completeness
See main article: Complete topological vector space.
Every topological vector space (and more generally, a topological group) has a canonical uniform structure, induced by its topology, which allows the notions of completeness and uniform continuity to be applied to it. If
is a metrizable TVS and
is a metric that defines
's topology, then its possible that
is complete as a TVS (i.e. relative to its uniformity) but the metric
is a
complete metric (such metrics exist even for
). Thus, if
is a TVS whose topology is induced by a pseudometric
then the notion of completeness of
(as a TVS) and the notion of completeness of the pseudometric space
are not always equivalent. The next theorem gives a condition for when they are equivalent:
If
is a closed vector subspace of a complete pseudometrizable TVS
then the quotient space
is complete. If
is a vector subspace of a metrizable TVS
and if the quotient space
is complete then so is
If
is not complete then
but not complete, vector subspace of
A Baire separable topological group is metrizable if and only if it is cosmic.[2]
Subsets and subsequences
- Let
be a separable locally convex metrizable topological vector space and let
be its completion. If
is a bounded subset of
then there exists a bounded subset
of
such that
S\subseteq\operatorname{cl}CR.
- Every totally bounded subset of a locally convex metrizable TVS
is contained in the closed convex balanced hull of some sequence in
that converges to
- In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.
- If
is a translation invariant metric on a vector space
then
for all
and every positive integer
- If
is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence
of positive real numbers diverging to
such that
- A subset of a complete metric space is closed if and only if it is complete. If a space
is not complete, then
is a closed subset of
that is not complete.
- If
is a metrizable locally convex TVS then for every bounded subset
of
there exists a bounded disk
in
such that
and both
and the auxiliary normed space
induce the same subspace topology on
Generalized series
As described in this article's section on generalized series, for any
-
indexed family family
of vectors from a TVS
it is possible to define their sum
as the limit of the
net of finite partial sums
F\in\operatorname{FiniteSubsets}(I)\mapstostyle\sum\limitsiri
where the domain
\operatorname{FiniteSubsets}(I)
is
directed by
If
and
for instance, then the generalized series
converges if and only if
converges unconditionally in the usual sense (which for real numbers,
is equivalent to
absolute convergence). If a generalized series
converges in a metrizable TVS, then the set
\left\{i\inI:ri ≠ 0\right\}
is necessarily
countable (that is, either finite or
countably infinite); in other words, all but at most countably many
will be zero and so this generalized series
style\sum\limitsiri~=~style\sum\limits\stackrel{i{ri ≠ 0}}ri
is actually a sum of at most countably many non-zero terms.
Linear maps
If
is a pseudometrizable TVS and
maps bounded subsets of
to bounded subsets of
then
is continuous. Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS. Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.
If
is a linear map between TVSs and
is metrizable then the following are equivalent:
-
is continuous;
-
is a (locally) bounded map (that is,
maps (von Neumann) bounded subsets of
to bounded subsets of
);
-
is sequentially continuous;
- the image under
of every null sequence in
is a bounded set where by definition, a is a sequence that converges to the origin.
-
maps null sequences to null sequences;
Open and almost open maps
Theorem: If
is a complete pseudometrizable TVS,
is a Hausdorff TVS, and
is a closed and
almost open linear surjection, then
is an open map.
Theorem: If
is a surjective linear operator from a
locally convex space
onto a
barrelled space
(e.g. every complete pseudometrizable space is barrelled) then
is
almost open.
Theorem: If
is a surjective linear operator from a TVS
onto a
Baire space
then
is almost open.
Theorem: Suppose
is a continuous linear operator from a complete pseudometrizable TVS
into a Hausdorff TVS
If the image of
is non-
meager in
then
is a surjective open map and
is a
complete metrizable space.
Hahn-Banach extension property
See main article: Hahn-Banach theorem.
A vector subspace
of a TVS
has
the extension property if any continuous linear functional on
can be extended to a continuous linear functional on
Say that a TVS
has the
Hahn-Banach extension property (
HBEP) if every vector subspace of
has the extension property.
The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:
If a vector space
has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.
Notes
Proofs
Bibliography
- Bourbaki. Nicolas. Nicolas Bourbaki. Annales de l'Institut Fourier. French. 0042609. 5–16 (1951). Sur certains espaces vectoriels topologiques. 2. 1950. 10.5802/aif.16. free.
- Book: Husain, Taqdir. Barrelledness in topological and ordered vector spaces. Springer-Verlag. Berlin New York. 1978. 3-540-09096-7. 4493665 .
Notes and References
- Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
- Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)