Everytime I look at a topology text, I get overwhelmed with the sheer density of the terminology. You've got first and second countable spaces, Hausdorff spaces, Tychonoff spaces, preregular spaces, completely Hausdorff spaces, Kolmogorov spaces, cofinite topologies, pseudometric spaces, uniform spaces, Zariski topology, regular spaces, paracompactness, metacompactness, orthocompactness, fully normal spaces, paranormal spaces, and so on, and so forth, ad practically infinitum.
Same thing goes for functional analysis: Asplund spaces, Frechet spaces, Baire spaces, Gateaux derivatives, Lipschitz functions, porous sets, meagre sets, $\Gamma$-null sets, locally convex topological vector spaces, absolutely convex sets, cones, nuclear sets, etc.
Now, in attempting to learn terminology on my own I usually go about it the way of "learning as much terminology in one sitting as I can" and then exploring the consequences of these ideas later on, as a whole.
But somehow, this feels "wrong". I feel like I should be taking it one step at a time, and fully exploring one idea (or at least, thoroughly understanding it) before jumping right to the next one.
But somehow that feels wrong as well, I feel like I'm missing core concepts when I don't know certain terms "a priori", or I don't have a sense of the big picture because I don't see where else an idea can lead, or how it relates to other concepts.
I hope this doesn't come off as two open-ended, I hadn't intended it to be, but I was simply wondering if there is an agreed upon "order" in which to learn things. What is the standard for learning terminology (lets keep it specific to topology and functional analysis, though if other fields are applicable then there is no reason not to mention them), if such a standard exists? Is it often better to explore concepts slowly, one at a time, or to gather as much "base level" terminology as you can?
Much of you listed is not relevant (as an introductory material). I am not sure what book are you reading (given what you wrote); for General Topology I suggest Munkres. Here is my list (somewhat biased since I am not an analyst):
General Topology:
A. Definitions related to spaces: Open, closed, closure, interior, compact, Hausdorff, connected, metric, complete, metric topology, order, order topology, subspace topology, quotient topology, product topology, weaker/stronger topology, 1st countable, 2nd countable topology.
B: Definitions related to maps between spaces: Continuous, homeomorphism, proper map, pointwise convergence, uniform convergence, equicontinuous, Lipschitz.
Functional Analysis:
A: Definitions related to spaces: Vector space, subspace, topological vector space, separable space, normed space, Banach space, Hilbert space, dual space, reflexive space, Frechet space, convex subset, convex cone.
B: Definitions related to maps between spaces: Linear operator and a linear functional, operator norm, bounded operator, compact operator, trace, Fredholm operator, nuclear operator, weak *-topology, operator algebra.