Is there possibilty to synthesise oxygen from carbondioxide?

There is definitely a possibility of doing this, and research towards making it real. You could look at Daniel Nocera's work at MIT, for instance.

Don't lose track of the requirement for lots of energy to do this, though. You aren't going to be separating oxygen from anything without pushing on it pretty hard.

Yes, Sandia Labs has the CR-5 reactor which heats a cobalt oxide ring till it gives off one oxygen atom releasing it to the atmosphere then they cool the cobalt oxide slightly and expose it to CO2 or H2O and it snatches the oxygen from the CO2 leaving CO gas or from the H2O leaving H2 gas. The net effect is that the oxygen from the CO2 or H2O is released into the atmosphere. The purpose of the reactor was originally to more efficiently extract H2 from H2O for a hydrogen economy but it was quickly realized that it also produced CO from CO2. This is important because the mixture of CO and H2 is called syngas and is used in refineries because it synthesizes into linear hydrocarbons in exothermic reactions. You can literally synthesize gasoline and diesel with syngas and indeed that's how we meet the new federal ultra low sulfur diesel requirements.

There are also several researchers working on artificial photosynthesis, some using carbon nanotubes.

The most likely method for space habitats would be to grow blue green algae in bioreactors.

However, it's not the lack of oxygen production that our imbalance comes from. The imbalance is that we are releasing hydrocarbons stored deep within the Earth and reintroducing it into our biosystem where we burn it with our oxygen producing CO2 and H2O. The imbalance is that this is carbon and hydrogen that was not in our atmosphere before we brought out the fossil fuels. The hydrogen isn't affecting us much but the CO2 is a greenhouse gas and affects our climate. So far, the proposed solution of industry has been CO2 sequestration, burying both the carbon and the oxygen when in reality it's the carbon and the hydrogen that should be buried. There are carbon sequestration methods that buries the carbon alone in soil as biochar which also increases soil fertility but they are largely ignored by industry.

Some. Not much for a couple of reasons. Reason number one: it won't do any good if we don't stop pumping gigatons of CO2 int the atmosphere.

More to the point, research isn't really necessary. Provided we do stop producing so much excess CO2, converting the existing excess is simple: plant more trees and stop cutting down the ones we have. This works, it has side benefits and it's cheap.

Look, we need research -- lots of it -- to get alternative energy technologies better developed to replace fossil fuels. But dealing with existing CO2? Sometimes the simple, low-tech solutions re the best. In this case, a guy with a shovel and a bag of seeds is all yo need

Haemoglobin in the blood has greatest affinity to oxygen adopted through other gases akin to carbondioxide, carbonmonoxide etc. It however has much less affinity towards different inert gases. Therefore when we inhale air which is nothing however a blend of gases, haemoglobin picks up oxygen and transports the identical to the exclusive parts of our body. The cells use these oxygen for performing quite a lot of functions and the byeproduct carbondioxide is carried again to the lungs and exhaled.

The plants do that. Do not cut the trees.

nope, sorry