Reddit reviews Kant's Critique of Pure Reason: An Introduction (Cambridge Introductions to Key Philosophical Texts)
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Well, in very short terms, Kant considered space (and time) to be a formal condition of human cognition. Formal in the sense that it is the constant, unchanging form of otherwise ever changing perceptions. However, space is not a property of things in themselves, in other words, things in themselves have no spatial (nor temporal) properties. Think of space and time in Kant as a framework we add to the world in order to make it comprehensible. Furthermore, Kant believes that space is a priori (but, particular) and his proof of space being a formal condition of human condition can be found in Transcendental Aesthetic at the beginning of CPR if I remember correctly. There you will find he argues among other things, there is only one space and we could imagine space without objects, but not objects without space.
Here is an example off the top of my head, mind you I don't know a whole lot about non-Euclidean (or Euclidean in fact) geometry. Now if there were to exist some sound non-Euclidean view of geometry, say for example Riemannian geometry, that would allow for say wormholes to exist and this account would be radically different to that of Euclidean or Newtonian (and in some respect Leibnizian) views of space Kant accepted, you can probably see how this would undermine his position if it were a favoured view. It would perhaps allow for four-dimensional space-time to be bent and loop back on itself, therefore not being as linear (in case of time) as Kant argued it is.
I personally feel Kant was heavily influenced by Euclidean geometry and Newton's theory of space. Especially Euclidean geometry which he believed to be complete and unquestionable (for the most part at least), so much that he used it often in describing and arguing his fundamental philosophical ideas such as those of a priori judgments and obviously transcendental character of space and time as formal conditions of our cognition. Any account of geometry that would prove Euclidean geometry wrong or simply insufficient could therefore possibly object to Kant's "system" as you understandably put it.
I hope this brief and very informal interpretation of mine will help you frame the problem and understand why it exists in the first place (provided I am not mistaken in my understanding of Kant, of course). However, I am afraid it would not help you much if you didn't already, or don't currently study Kant, in which case I would direct you to Jill Vance Buroker's Introduction, Korner's Kant, and Kant's Critique of Pure Reason and Prolegomena. Links provided are of purely informative nature so you could see if your department's library has a copy or find them online; both have chapters or sections devoted to time and space, but I would advise reading the preceding chapters on Kant's critical project if you are not already acquainted with it. They may look daunting, but what you seem to need them for should not take up more than fifty pages in each.
Kant's Critique of Pure Reason (CPR) gets taught very rarely in this department. The department recognizes the need to have a course on Kant's CPR (or, alternatively, on Kant's shorter version of the CPR, his "Prolegomena to Any Future Metaphysics"), but the Maryland philosophy department (a) doesn't have many faculty who work on the history of philosophy, and (b) those faculty who do work in the history of philosophy either do work on ancient philosophy (Rachel Singpurwalla, Quinn Harr, Kelsey Gipe) or on Spinoza and other historical Jewish philosophers (Charles Manekin).
Sam Kerstein of course does work on historical Kant, but Sam's focus and interests in Kant is fairly exclusively directed towards Kant's moral philosophy. This is why Sam teaches a 400-level class on Kant's Groundwork every other year or so.
The upshot is that I am the first person to teach a course on Kant's CPR at this department in many years (6+). I'll probably teach the course again either next school year or, if not next year, then the following year. Unfortunately, that sounds like it might be too late for you (from what you've said, it sounds like you graduate this year).
Fortunately, I would argue that it is better for you to have taken a class on Kant's Groundwork before you graduate than Kant's CPR. Kant's ethics is more important to contemporary philosophy than his epistemology and metaphysics. That being said, I do hope you decide to give the CPR a read on your own time someday or at least read a secondary source on Kant that covers the important content from the CPR in detail.
If you decide to read Kant's CPR on your own, let me recommend some resources. First, I'd suggest you watch the following two videos about Hume and the following three videos on Kant as background (although, unfortunately there isn't a video connecting Kant to Hume through how Kant's CPR is in large part a response to Hume's skepticism):
You might want to also read one general secondary source on Kant's whole philosophical system before the CPR. Here are a few suggestions to pick one from - all accessible to a upper-level undergraduate student or graduate student. I am not sure which one would be best to recommend:
There are also several specific secondary sources aimed at upper-level undergraduates and graduate students that specifically cover Kant's CPR. One of these might be useful to have besides you as you read the CPR for those times you get lost:
Kant's a hard read due to the structured state of his language and how this translates. Harder than most.
If you really wish to jump in, I recommend the following from the various attempts that I've taken at reading and understanding all of it properly:
http://www.amazon.com/Kants-Critique-Pure-Reason-Introductions/dp/0521618258
http://www.amazon.com/Kant-Metaphysics-Cambridge-History-Philosophy/dp/0521566738
http://www.amazon.com/Foundations-Metaphysics-Morals-Immanuel-Kant/dp/0023078251