I'm reading an article of topology and i came across a Properties :
Properties :
Closed subspaces and arbitrary products of Čech-complete spaces are Čech-complete
Every Čech-complete space is a Baire space
Where the author just explain the concept. I am stuck on what actually does it mean and how to prove it. Is there an easy way to prove this propeties?
Just read the relevant part (paragraph 3.9) of Engelking's book General Topology (1989 2nd ed.) The proof of the second fact is not hard and resembles the proof that locally compact Hausdorff spaces are Baire. Or show generally that a dense $G_\delta$ of a Baire space is Baire too, which is easy from definitions.
Fact 1. is false as stated, as $\Bbb N^{\aleph_1}$ is not Čech-complete and $\Bbb N$ is. It does hold for countable products and the proof is not hard from the definition (3.9.8 in Engelking). The closed subspace part is true, but needs another characterisation of Čech-complete spaces in terms of closed subsets with the FIP that Engelking shows in 3.9.2.
The author of the paper just assumes that readers have access to standard works like Engelking (or the Handbook of Set Theoretic topology or the Encyclopedia of General Topology) so they know how to look up such facts or find references for them. You should check them (especially if the notions are new) and here there even is a mistake in one of them...