If a set is equipped with a dense, complete order then the corresponding topological space is connected - and hence, so are its continuous images, even in unordered spaces. What happens if we remove the completeness condition on the order, and consider a general densely ordered space X?

I know that f is continuous from X to another ordered space, then the induced order on f(X) need not be dense. But is there some weaker property of densely ordered spaces which is preserved under continuous maps? If so, can such a property be generalized to unordered spaces in the same sort of way that "densely and completely ordered" generalizes to "connected"?

Edit: as discussed below, here is the followup question asked by Noah, as well as another one added by me.

If a set is equipped with a dense, complete order then the corresponding topological space is connected - and hence, so are its continuous images, even in unordered spaces. What happens if we remove the completeness condition on the order, and consider a general densely ordered space X?

I know that f is continuous from X to another ordered space, then the induced order on f(X) need not be dense. But is there some weaker property of densely ordered spaces which is preserved under continuous maps? If so, can such a property be generalized to unordered spaces in the same sort of way that "densely and completely ordered" generalizes to "connected"?

Edit: as discussed below, here is the followup question asked by Noah, as well as another one added by me.

If a set is equipped with a dense, complete order then the corresponding topological space is connected - and hence, so are its continuous images, even in unordered spaces. What happens if we remove the completeness condition on the order, and consider a general densely ordered space X?

I know that f is continuous from X to another ordered space, then the induced order on f(X) need not be dense. But is there some weaker property of densely ordered spaces which is preserved under continuous maps? If so, can such a property be generalized to unordered spaces in the same sort of way that "densely and completely ordered" generalizes to "connected"?

If a set is equipped with a dense, complete order then the corresponding topological space is connected - and hence, so are its continuous images, even in unordered spaces. What happens if we remove the completeness condition on the order, and consider a general densely ordered space X?

I know that f is continuous from X to another ordered space, then the induced order on f(X) need not be dense. But is there some weaker property of densely ordered spaces which is preserved under continuous maps? If so, can such a property be generalized to unordered spaces in the same sort of way that "densely and completely ordered" generalizes to "connected"?

Edit: as discussed below, here is the followup question asked by Noah, as well as another one added by me.