Is the Component of a Metric Space Always Open or Closed?

[jessicaw]

Is component(maximal connected set) of a metric space open or closed or both(clospen)?or even can be half-open(not open and not closed)?
I know it is a silly question as (3,5] is a component in R,right?
However some theorem i encountered stated that component must be closed or must be open. I know they can't be contradictory but i need help in understanding this. Thx~

 

[Eynstone]

A connected component is always closed, but may not be open.

 

[jessicaw]

A connected component is always closed, but may not be open.

BUT (3,5] is not closed on R, right?

 

[micromass]

No, it is not closed. But it's not a component to. R itself is connected, thus R itself is the component.

 

[jessicaw]

so interval in R1 is in the form [x,y]?

 

[micromass]

I'm not sure what you mean...

Every interval (wether it is [a,b], [a,b[, ]a,b], ...) is connected in R. But they are not components, since they are not MAXIMAL connected sets. Indeed, R itself is connected and is thus the maximal connected set. Thus R itself is a component. The intervals are not components, but are connected...