topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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
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
continuous metric space valued function on compact metric space is uniformly continuous
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
Analysis Theorems
A continuous map $f : X \to Y$ between topological spaces is called a local homeomorphism if restricted to a neighbourhood of every point in its domain it becomes a homeomorphism onto its image which is required to be open.
One also says that this exhibits $X$ as an étale space over $Y$.
Notice that, despite the similarity of terms, local homeomorphisms are, in general, not local isomorphisms in any natural way. See the examples below.
A local homeomorphism is a continuous map $p \colon E \to B$ between topological spaces (a morphism in Top) such that
or equivalently
See also at étale space.
For $X$ any topological space and for $S$ any set regarded as a discrete space, the projection
is a local homeomorphism.
For $\{U_i \to Y\}$ an open cover, let
be the disjoint union space of all the pathches. Equipped with the canonical projection
this is a local homeomorphism.
In general, for every sheaf $A$ of sets on $Y$; there is a local homeomorphism $X \to Y$ such that over any open $U \hookrightarrow X$ the set $A(U)$ is naturally identified with the set of sections of $Y \to X$. See étale space for more on this.
A local homeomorphism is an open map.
Let $f \colon X \to Y$ be a local homeomorphism and $U \subset X$ an open subset. We need to see that the image $f(U) \subset Y$ is an open subset of $Y$. For this we may equivalently show that each $y \in f(U)$ has an open neighbourhood inside $f(U)$.
But since any function is surjective onto its image, there exists $x \in U$ with $f(x) = y$. By local homeomorphy of $f$, this $x \in X$ has an open neighbourhood $U_x \subset X$ with $f_{|U_x} \colon U_x \to f(U_x)$ a homeomorphism. Since $U \cap U_x \subset U_x$ is an open neighbourhood of $x$ in $U_x$, the homeomorphy of $f_{|U_x}$ implies that $f(U \cap U_x) \subset f(U)$ is an open neighbourhood of $f(x) = y$.
Last revised on June 30, 2022 at 06:05:19. See the history of this page for a list of all contributions to it.