Especially when you study algebraic topology, you'll encounter a lot of casually stated claims, involving homeomorphisms, quotient space constructions, or deformation retracts. One very famous example would be the identification of $S^n$ as the quotient of $D^n$ by its boundary $S^{n-1}$. Some people may well accept this fact simply by imagining the case $n=2$ where a disc 'wraps up' to become a sphere, and then by hand-waving. More cautious people might look for an explicit formula for a homeomorphism, which is not so difficult in this example.
However, as I am not even close to a genius topologist, I occasionally face topological claims which are neither easily visualized in mind (at a first glance) nor easily proved by a detailed formula. For instance, the following examples in Hatcher's Algebraic Topology do not resonate with my (inborn) intuition.
$\cdot~ \mathbb{R}^3 - S^1$ is homeomorphic to a $\epsilon$-neighborhood of $S^1 \vee S^2$ embedded in $\mathbb{R}^3$.
$\cdot~$ An orientable surface of genus $g$ is obtained by identifying pairs of edges from a polygon with $4g$ sides.
$\cdot~$ Given a 2-simplex $[v_0, v_1, v_2]$, identifying the edges $[v_0,v_1]\sim [v_1,v_2]$ produces a Mobius band.
Please note that I am not asking for visualization or formal proofs for these specific examples. To me they are somewhat justified but not perfectly. That is, I do have some ways to explain each of these to myself, but I do not consider them as satisfying as a well-written proof in analysis. However, I also don't want to spend much time finding a rigorous argument each time I come across something like these. Believing that I am not the only person in the world feeling this sort of awkward dissatisfaction, I would be very happy to learn his/her attitude or opinion on my worries from someone who had gone through a similar problem.
Edit2: As Theoretical Economist provided a better paraphrase of the question, I would like to include it. Would there be a strategy to justify non-intuitive topological claims in a mathematically satisfactory (or acceptable) way, but not writing down all the tedious details? I think there should be some common way of reasoning, considering how many topological statements there are which everyone agrees with but no one actually writes the proof down.
Edit: So my question is, to those who are experienced in topology: Haven't you ever felt uncomfortable with some casual arguments because they weren't intuitively so clear to you? In cases when you felt so, did you always bother to 'rigorously' prove the argument? I am asking this because I am not good at quickly understanding or coming up with intuitive topogical imagination, and also believe that it does not significantly hinder studying and understanding core ideas of AT, but it still disturbs me sometimes.
Thanks in advance!