Vipul: Created page with "==Definition== A topological space is termed '''almost discrete''' if it satisfies the following equivalent conditions: # Every open subset is a closed subset # ..."

2012-01-27T04:12:57Z

Created page with "==Definition== A topological space is termed '''almost discrete''' if it satisfies the following equivalent conditions: # Every open subset is a closed subset # ..."

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==Definition==

A [[topological space]] is termed '''almost discrete''' if it satisfies the following equivalent conditions:

# Every [[open subset]] is a [[closed subset]]
# Every [[closed subset]] is an [[open subset]]
# It is both an [[defining ingredient::Alexandrov space]] (i.e., it is finitely generated) and a [[defining ingredient::zero-dimensional space]] (i.e., it has a [[basis]] of [[clopen subset]]s).

{{topospace property}}
==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::discrete space]] || || || || {{intermediate notions short|almost discrete space|discrete space}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::Alexandrov space]] || || || ||
|-
| [[Stronger than::zero-dimensional space]] || || || ||
|}

Almost discrete space - Revision history

Vipul: Created page with "==Definition== A topological space is termed '''almost discrete''' if it satisfies the following equivalent conditions: # Every open subset is a closed subset # ..."