see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
fiber space, space attachment
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
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
Theorems
Given a continuous function between two connected closed oriented manifolds of the same dimension, its degree is a measure for how often the function “wraps its domain around its codomain”.
For $X$ is a connected closed oriented manifold of dimension $n$, its top homology group $H_n(X) = H_n(X; \mathbb{Z})$ is isomorphic to $\mathbb{Z}$, where the generator $1 \in \mathbb{Z}$ is identified with the orientation class $[\omega_X]$ of $X$, the fundamental class of $X$.
Given a continuous map $f \colon X \to Y$ between two such manifolds, the homomorphism $f_\ast = H_n(f) \colon H_n(X) \to H_n(Y)$ is therefore specified by the integer $n$ such that $f_\ast [\omega_X] = n [\omega_Y]$. This integer is called the degree of $f$.
We suppose throughout that $X$ and $Y$ are connected closed oriented manifolds of the same dimension $n$. The degree of a continuous function $g \colon X \to Y$ is frequently computed according to the following considerations:
The space of continuous functions $g \colon X \to Y$ has a dense subspace consisting of smooth functions $f \colon X \to Y$, and in particular every continuous function $g$ is homotopic to a smooth function $f$. It therefore suffices to compute the degree of $f$.
By Sard's theorem, the set of singular values? of a smooth function $f$ has measure zero (using for example the orientation on $Y$ to define a volume form and hence a measure). Accordingly, we may choose a regular value? $y \in Y$.
The inverse image $f^{-1}(y)$ is a compact $0$-dimensional manifold, hence consists of finitely many (possibly zero) points $x_1, \ldots, x_r \in X$. Since these are regular points, $f$ restricts to a diffeomorphism
where $U_i$ is a small neighborhood of $x_i$ and $V$ is a small neighborhood of $y$. The diffeomorphism $f_i$ either preserves or reverses the orientation of $U_i$, i.e., the sign of the determinant as a mapping between differential n-forms
is either $+1$ or $-1$.
By a straightforward application of the excision axiom? in homology, it follows that the degree of $f$ is the sum of these signs:
and this quantity is independent of the choice of regular value $y$.