Neighbourhood retract

This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces

Definition

Definition with symbols

A subspace $A$ of a topological space $X$ is termed a neighbourhood retract in $X$ is there is an open subset $U$ of $X$ containing $A$, such that $A$ is a retract of $U$.

Formalisms

In terms of the neighbourhood operator

This property is obtained by applying the neighbourhood operator to the property: retract

Relation with other properties

Stronger properties

• Strong deformation retract
• Homotopy retract
• Retract
• Open subset

Metaproperties

Transitivity

This property of subspaces of topological spaces is transitive. In other words, if $A$ satisfies the property as a subspace of $B$ and $B$ satisfies the property as a subspace of $C$ then $A$ satisfies the property as a subspace of $C$

If $A$ is a neighbourhood retract of $B$ and $B$ is a neighbourhood retract of $C$, then we can find a neighbourhood of $A$ in $C$, for which it is a retract.