Topology (topology v0.1.4)
Returns a topology of the given set.
Link to this section Summary
Functions
Returns a closed set system of the given topological space.
Returns a discrete space of the given set.
Returns a discreate topology of the given set.
Returns a discrete space of the given set.
Returns a indiscreate topology of the given set.
Returns a boolean indicating whether the given set is an closed set of the given topological space.
Returns a boolean indicating whether the given set is an open set of the given topological space.
Returns a topological spaces of the given set.
Returns a topology of the given set.
Link to this section Types
topological_space()
@type topological_space() :: {underlying_set(), topology()}
topology()
@type topology() :: MapSet.t()
underlying_set()
@type underlying_set() :: MapSet.t()
Link to this section Functions
closed_set_system(arg)
@spec closed_set_system({underlying_set(), topology()}) :: MapSet.t()
Returns a closed set system of the given topological space.
examples
Examples
iex> topological_space =
...> {
...> MapSet.new([:a, :b, :c]),
...> MapSet.new([
...> MapSet.new([]),
...> MapSet.new([:b]),
...> MapSet.new([:a, :b]),
...> MapSet.new([:a, :b, :c])
...> ])
...> }
iex> Topology.closed_set_system(topological_space)
MapSet.new([
MapSet.new([:a, :b, :c]),
MapSet.new([:a, :c]),
MapSet.new([:c]),
MapSet.new([]),
])
discrete_space(set)
@spec discrete_space(MapSet.t()) :: topological_space()
Returns a discrete space of the given set.
examples
Examples
iex> Topology.discrete_space(MapSet.new([:a, :b]))
{MapSet.new([:a, :b]), MapSet.new([MapSet.new([]), MapSet.new([:a]), MapSet.new([:b]), MapSet.new([:a, :b])])}
discrete_topology(set)
Returns a discreate topology of the given set.
examples
Examples
iex> Topology.discrete_topology(MapSet.new([:a, :b]))
MapSet.new([MapSet.new([]), MapSet.new([:a]), MapSet.new([:b]), MapSet.new([:a, :b])])
indiscrete_space(set)
@spec indiscrete_space(MapSet.t()) :: topological_space()
Returns a discrete space of the given set.
examples
Examples
iex> Topology.indiscrete_space(MapSet.new([:a, :b]))
{MapSet.new([:a, :b]), MapSet.new([MapSet.new([]), MapSet.new([:a, :b])])}
indiscrete_topology(set)
Returns a indiscreate topology of the given set.
examples
Examples
iex> Topology.indiscrete_topology(MapSet.new([:a, :b]))
MapSet.new([MapSet.new(), MapSet.new([:a, :b])])
is_closed_set?(set, arg)
@spec is_closed_set?( MapSet.t(), {underlying_set(), topology()} ) :: boolean()
Returns a boolean indicating whether the given set is an closed set of the given topological space.
examples
Examples
iex> topological_space =
...> {
...> MapSet.new([:a, :b, :c]),
...> MapSet.new([
...> MapSet.new([]),
...> MapSet.new([:b]),
...> MapSet.new([:a, :b]),
...> MapSet.new([:a, :b, :c])
...> ])
...> }
iex> Topology.is_closed_set?(MapSet.new([:c]), topological_space)
true
is_open_set?(set, arg)
@spec is_open_set?( MapSet.t(), {underlying_set(), topology()} ) :: boolean()
Returns a boolean indicating whether the given set is an open set of the given topological space.
examples
Examples
iex> topological_space =
...> {
...> MapSet.new([:a, :b, :c]),
...> MapSet.new([
...> MapSet.new([]),
...> MapSet.new([:b]),
...> MapSet.new([:a, :b]),
...> MapSet.new([:a, :b, :c])
...> ])
...> }
iex> Topology.is_open_set?(MapSet.new([:a, :b]), topological_space)
true
open_set_systems(set)
topological_spaces(set)
Returns a topological spaces of the given set.
examples
Examples
iex> Topology.topological_spaces(MapSet.new([:a, :b]))
[
{MapSet.new([:a, :b]), MapSet.new([MapSet.new([]), MapSet.new([:a, :b])])},
{MapSet.new([:a, :b]), MapSet.new([MapSet.new([]), MapSet.new([:a]), MapSet.new([:a, :b])])},
{MapSet.new([:a, :b]), MapSet.new([MapSet.new([]), MapSet.new([:b]), MapSet.new([:a, :b])])},
{MapSet.new([:a, :b]), MapSet.new([MapSet.new([]), MapSet.new([:a]), MapSet.new([:b]), MapSet.new([:a, :b])])}
]
topologies(set)
Returns a topology of the given set.
examples
Examples
iex> Topology.topologies(MapSet.new([:a, :b]))
MapSet.new([
MapSet.new([MapSet.new([]), MapSet.new([:a, :b])]),
MapSet.new([MapSet.new([]), MapSet.new([:a]), MapSet.new([:a, :b])]),
MapSet.new([MapSet.new([]), MapSet.new([:b]), MapSet.new([:a, :b])]),
MapSet.new([
MapSet.new([]),
MapSet.new([:a]),
MapSet.new([:b]),
MapSet.new([:a, :b])
])
])