Hereditary Properties of Topological Spaces

# Hereditary Properties of Topological Spaces

Recall from the Topological Subspaces that if $(X, \tau)$ is a topological space and $A \subseteq X$ then the subspace topology $\tau_A$ on $A$ is given as the collection of intersections of all open sets in $X$ with $A$, that is:

(1)
\begin{align} \quad \tau_A = \{ A \cap U : U \in \tau \} \end{align}

We proved that $\tau_A$ did indeed form a topology on $A$ since $\tau_A$ is the initial topology induced by the inclusion map $i : A \to X$ (the coarsest topology on $A$ making $i$ continuous) on $A$ where $i(a) = a$ for all $a \in A$.

We will now look at these topological subspaces a little more thoroughly. We first begin with classifying properties of $(X, \tau)$ that are "passed down" to $(A, \tau_A)$.

 Definition: Let $(X, \tau)$ be a topological space. A property of $(X, \tau)$ is said to be Hereditary if for all $A \subseteq X$ we have that the subspace $(A, \tau_A)$ also has that property. A property of $X$ that is not hereditary is said to be Nonhereditary.

In other words, a property on $X$ is hereditary if every subspace of $X$ with the subspace topology also has that property.

We will prove that first and second countability are hereditary properties on the following pages:

Of course not all properties of $X$ are hereditary. We will show that separability is a nonhereditary property on the following page: