Manifolds: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF?

Question

Let me denote the following

• (TOP) topological manifolds
• (PDIFF), for piecewise differentiable
• (PL) piecewise-smooth manifolds
• (DIFF) the smooth manifolds
• (TRI) triangulable manifolds

Is it true that

(1) TOP $\supseteq$ TRI ?

Namely, every TRI must be TOP manifolds?

(2) TRI $\supseteq$ PL ?

Namely, every PL must be TRI manifolds?

(3) TRI $\supseteq$ DIFF ?

Namely, every DIFF must be TRI manifolds?

(4) PL $\supseteq$ DIFF ?

Namely, every DIFF must be PL manifolds?

(5) So in a short summary, is it true that

$$\text{ TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF} ?$$

How to think these categories of manifolds in terms of categories (or sets with whose objects are manifolds)? What are their intersections, unions and complements?

Answer 1

Yes, these inclusions are all true.

1) is true because "manifold" and "topological manifold" are actually synonyms.

2) is true by an almost obvious construction: very roughly speaking, a locally finite covering by charts in the given PL atlas can be triangulated one chart at a time.

4) is true by a theorem of Whitney.

3) follows from 2) and 4).