see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
subsets are closed in a closed subspace precisely if they are closed in the ambient space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Basic homotopy theory
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
The Whitney $C^k$ topology is a topology on the set $C^\infty(X,Y)$ of smooth functions between two smooth manifolds, induced from the natural topology of the order-$k$ jet spaces of maps from $X$ to $Y$.
Often, and espcially for $k = \infty$ this is just called the $C^k$-topology .
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