TITLE: How can you describe topology to a non-mathematician without using continuous deformations?
QUESTION [12 upvotes]: One of the most frequently used ways to describe topology to non-mathematicians is that it studies the properties of objects that are preserved under deformations where ripping or tearing is not allowed.
I like this definition, but I've realized recently that it gives the wrong idea. In fact, this describes isotopy instead of homeomorphism.
Does it matter? Take an annulus, cut it into a strip of paper. Twist it once and glue, then you get a Möbius strip. Twist is twice and glue, what you get is another annulus.
But there is no way to deform a standard annulus into a doubly twisted annulus. So the intuition is very wrong here.
In addition topology is much bigger than manifolds and isotopies anyways. So my question is:

What is an effective way to describe topology that will give an intuitive understanding of continuity, connectedness, and compactness, without using the idea of smooth deformations?

REPLY [2 votes]: It's kind of uncanny that a friend who's majoring in biology asked me this question just now.
For me, I think it's best to start with the concept of basis for a topology on a set $X$. 
A basis is best described as building blocks of a space.
I think it's kind of difficult to describe a continuous function for a general topological space to a non-mathematician. However, maybe a subset of it, metric spaces, we can describe continuous functions as maps that preserves distances between points.
For connectedness, we can describe it as a property of a space that cannot be split into any two building blocks or unions of building blocks. (The concept of only the empty set and the whole space being clopen)
For compactness, I think it's best described as being able to find finitely many building blocks that can eclipse the space. But then again, the definition of finite may differ across non-mathematicians. Or maybe you can try asking the questions: "What do you think being compact is? What's the dictionary definition of compact?" Maybe all these will stir some form of intuition.