A map is a function. A function is a map. Sometimes people use the word "map" in a special sense, to mean only a continuous function or only a smooth function. For you, we'll try to always put an adjective before "map".
Topological space: any type of space you can imagine. When speaking of it as a topological space, you are forbidden to talk about any of the non-topological properties of the space. Formal definition: a topological space consists of a set X, whose elements are called points, together with a collection of subsets of X, called open sets, satisfying three conditions (the empty set and the set X are both open, the intersection of any two open sets is open, and the union of any collection of open sets is open). A topological space can also be defined in terms of closed sets, interior operators, neighbourhoods, and other different but equivalent ways.
Topological property: a spatial property that depends purely on continuity, and not on differentiability, or distance, or parallelism, or angles, or any of that stuff we visualize so readily. Examples of topological properties: «closed curve C in the plane doesn't intersect itself», «point p is inside closed curve C that doesn't intersect itself». Examples of non-topological properties: «curve C has a corner» (uses differentiability); «p is the center of circle C» (uses distance); «vector AB has the same direction as vector CD» (uses parallelism).
Continuous map: a map from one topological space to another which respects topological properties; it does not tear. Formal definition: a function f: X -> Y is continuous at p iff the following holds: for every open set E containing f(p) there is an open set D containing x such that: x is in D implies f(x) is in E. (Here E is an open set in Y, and D is an open set in X.)
Homeomorphism: a reversible mapping between two topological spaces that preserves all topological properties. Formal math definition: a one-to-one and onto (same as reversible) continuous function from one topological space to another whose inverse is also continuous.
Embedding: a map from one topological space (X) to another (Y) which fits X completely into Y without squishing it. Formal math definition: a map f: X -> Y which is a homeomorphism from X to the image of f in Y but not necessarily a homeomorphism from X to all of Y.
Homotopy: 2 continuous maps between the same topological spaces are homotopic iff they can be continuously deformed into each other. The homotopy is the continuous deformation of one of these maps into the other.
Qualified homotopy: eg 2 smooth maps between the same smooth manifolds are smoothly homotopic iff they can be smoothly deformed into each other.
Topological manifold: just a special kind of topological space. It is required that every point on a topological manifold have a little patch (neighbourhood) around it that is homeomorphic to a little patch around 0 in Euclidean nspace (Rn), for some n. This n is the dimension of the manifold at that point. (It turns out that a manifold with 1 component must have the same dimension everywhere, so one usually speaks of the dimension of the manifold, full stop.)
Differential manifold: a topological manifold that is endowed with calculus-like notions (differential-manifold properties), so we're allowed to talk about maps being differentiable, or k-times differentiable, or even infinitely differentiable (aka smooth). But we're still not allowed to talk about distance, or angles, or parallelism.
Diffeomorphism: a reversible mapping between two differential manifolds that preserves all differential-manifold properties. Formal math definition: a one-to-one and onto (same as reversible) smooth function from one differential manifold to another whose inverse is also smooth.
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