# Thread: Topology in Complex Space?

1. ## Topology in Complex Space?

Was just wondering can you extend topology to complex spaces? So is the complex space a metric space with a metric that satisfies triangle inequality, symmetry, and positive definiteness?

2. Originally Posted by shilz222
Was just wondering can you extend topology to complex spaces? So is the complex space a metric space with a metric that satisfies triangle inequality, symmetry, and positive definiteness?
Roughly complex spaces can be thought of as real spaces with twice the number of dimensions.

RonL

3. Originally Posted by shilz222
Was just wondering can you extend topology to complex spaces? So is the complex space a metric space with a metric that satisfies triangle inequality, symmetry, and positive definiteness?
The topology in $\mathbb{C}$ can be tought as $\mathbb{R}^2$ (as CaptainBlank says) with the metric $d((x_1,x_2),(y_1,y_2)) = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$.

4. If any on any space there is a distance function defined, then that space is a metric space. The function $d(z,w) = \left| {z - w} \right| = \sqrt {\left( {{\mathop{\rm Re}\nolimits} (z) - {\mathop{\rm Re}\nolimits} (w)} \right)^2 + \left( {{\mathop{\rm Im}\nolimits} (z) - {\mathop{\rm Im}\nolimits} (w)} \right)^2 }$ is clearly a metric.