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.

$\blacksquare$