# Local Connectedness is not Preserved under Infinite Product

Jump to navigation
Jump to search

## Theorem

The property of local connectedness is not preserved under the operation of forming an infinite product space.

## Proof

Let $T = \struct {\CC, \tau_d}$ be the Cantor space.

Let $A_n = \struct {\set {0, 2}, \tau_n}$ be the discrete space of the two points $0$ and $2$.

Let $\ds A = \prod_{n \mathop = 1}^\infty A_n$.

Let $T' = \struct {A, \tau}$ be the product space where $\tau$ is the product topology on $A$.

From Cantor Space as Countably Infinite Product, $T'$ is homeomorphic to $T$.

From Totally Disconnected and Locally Connected Space is Discrete, we have that $A_n$ is locally connected.

But we also have that the Cantor Space is not Locally Connected.

Hence the result.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $29$. The Cantor Set: $9$