Discrete space: Difference between revisions
| (4 intermediate revisions by the same user not shown) | |||
| Line 7: | Line 7: | ||
A '''discrete space''' is a [[topological space]] satisfying the following equivalent conditions: | A '''discrete space''' is a [[topological space]] satisfying the following equivalent conditions: | ||
# It has a [[basis]] of [[open subset]]s comprising all the singleton subsets | |||
# Every singleton subset is an [[open subset]] | |||
# Every subset is an [[open subset]] | |||
# Every subset is a [[closed subset]] | |||
# Every subset is a [[clopen subset]] | |||
Given any set, there is a ''unique'' [[topology]] on it making it into discrete space. This is termed the ''discrete topology''. The discrete topology on a set is the [[finer topology|finest possible topology]] on the set. | |||
==Relation with other properties== | ==Relation with other properties== | ||
| Line 18: | Line 21: | ||
{| class="sortable" border="1" | {| class="sortable" border="1" | ||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
|- | |||
! Disconnectedness type | |||
|- | |- | ||
| [[Stronger than::totally disconnected space]] || the only connected subsets are singleton subsets || || || | | [[Stronger than::totally disconnected space]] || the only connected subsets are singleton subsets || || || | ||
|- | |- | ||
| [[Stronger than::door space]] || | | [[Stronger than::door space]] || every subset is open or closed || || || | ||
|- | |||
| [[Stronger than::submaximal space]] || every subset is locally closed || || || | |||
|- | |||
| [[Stronger than::weakly submaximal space]] || every finite subset is locally closed || || || | |||
|- | |||
| [[Stronger than::zero-dimensional space]] || has a [[basis]] of [[clopen subset]]s || || || | |||
|- | |||
| [[Stronger than::Alexandrov space]] ||arbitrary intersection of open subsets is open || || || | |||
|- | |||
| [[Stronger than::almost discrete space]] || Alexandrov and zero-dimensional || || || | |||
|- | |||
| [[Stronger than::extremally disconnected space]] || every [[regular open subset]] is closed || || || | |||
|- | |||
! Separation type | |||
|- | |- | ||
| [[Stronger than::locally compact space]] || || || || | | [[Stronger than::locally compact space]] || || || || | ||
| Line 34: | Line 53: | ||
|- | |- | ||
| [[Stronger than::regular space]] || || || || | | [[Stronger than::regular space]] || || || || | ||
|- | |||
! Extra structure type | |||
|- | |- | ||
| [[Stronger than::metrizable space]]|| || || || | | [[Stronger than::metrizable space]]|| || || || | ||
Latest revision as of 18:15, 28 January 2012
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is an opposite of compactness
Definition
A discrete space is a topological space satisfying the following equivalent conditions:
- It has a basis of open subsets comprising all the singleton subsets
- Every singleton subset is an open subset
- Every subset is an open subset
- Every subset is a closed subset
- Every subset is a clopen subset
Given any set, there is a unique topology on it making it into discrete space. This is termed the discrete topology. The discrete topology on a set is the finest possible topology on the set.
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Disconnectedness type | ||||
| totally disconnected space | the only connected subsets are singleton subsets | |||
| door space | every subset is open or closed | |||
| submaximal space | every subset is locally closed | |||
| weakly submaximal space | every finite subset is locally closed | |||
| zero-dimensional space | has a basis of clopen subsets | |||
| Alexandrov space | arbitrary intersection of open subsets is open | |||
| almost discrete space | Alexandrov and zero-dimensional | |||
| extremally disconnected space | every regular open subset is closed | |||
| Separation type | ||||
| locally compact space | ||||
| perfectly normal space | ||||
| completely normal space | ||||
| monotonically normal space | ||||
| completely regular space | ||||
| regular space | ||||
| Extra structure type | ||||
| metrizable space | ||||
| CW-space | ||||
| polyhedron |
Related properties
Compactness is the opposite of discreteness in some sense. The only topological spaces that are both discrete and compact are the finite spaces.
Metaproperties
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
A (finite?) direct product of discrete spaces is discrete.
Hereditariness
This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces
Any subspace of a discrete space is discrete under the induced topology.