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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

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Approximations by compact sub-spaces

Q1. Consider the space of rationals $\mathbb{Q}$. It is the direct limit of the family of finite unions of convergent sequences (including their limits), ordered by inclusion, as a set is closed iff i …
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5 votes
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One-dimensional compacta as projective limits

References to Engelking's Dimension Theory (1978) ISBN 0-444-85176-3. The answer is yes for compact metrizable spaces, see Section 1.13. In general it is no in general, see Example 3.3.8 (Lokucievski …
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