Topological Vector Space over Connected Topological Field is Connected
Theorem
Let $K$ be a connected topological field.
Let $X$ be a topological vector space over $K$.
Then $X$ is connected.
Proof
From the definition of a topological vector space, the mapping $\circ_X : K \times X \to X$ defined by:
- $\map {\circ_X} {\lambda, x} = \lambda x$
for $\tuple {\lambda, x} \in K \times X$ is continuous.
Let $x \in X$.
From Horizontal Section of Continuous Function is Continuous, the mapping $c_x : K \to X$ defined by:
- $\map {c_x} \lambda = \lambda x$
for $\lambda \in K$ is continuous.
Since $K$ is connected, we have that:
- $c_x \sqbrk K$ is connected
from Continuous Image of Connected Space is Connected.
That is:
- $K x$ is connected.
Since $x \in K x$ for each $x \in X$, we have:
- $\ds X = \bigcup_{x \in X} K x$
Since $0_K \circ x = {\mathbf 0}_X$ for each $x \in X$, we have that:
- $\ds {\mathbf 0}_X \in \bigcap_{x \in X} K x$
From Union of Connected Sets with Common Point is Connected, we have that:
- $\ds \bigcup_{x \in X} K x$ is connected.
Hence $X$ is connected.
$\blacksquare$