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
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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The definition of the concept of sub-nets of a net requires some care. The point of the definition is to ensure that compact spaces are equivalently those for which every net has a converging subnet.
There are several different definitions of ‘subnet’ in the literature, all of which intend to generalise the concept of subsequences. We state them now in order of increasing generality. Note that it is Definition 3 which is correct in that it corresponds precisely to refinement of filters. However, the other two definitions (def. 1, def. 2) are sufficient (in a sense made precise by theorem 1 below) and may be easier to work with.
(Willard, 1970).
Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a subnet of $x$ if:
We have a function $f\colon B \to A$ such that
(Kelley, 1955).
Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a subnet of $x$ if:
We have a function $f\colon B \to A$ such that
Notice that the function $f$ in definitions 1 and 2 is not required to be an injection, and it need not be. As a result, a sequence regarded as a net in general has more sub-nets than it has sub-sequences.
(Smiley, 1957; Årnes & Andenæs, 1972).
Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a subnet of $x$ if:
The eventuality filter of $y$ refines the eventuality filter of $x$. (Explicitly, for every $\alpha \in A$ there is a $\beta \in B$ such that, for every $\beta_1 \geq \beta \in B$ there is an $\alpha_1 \geq \alpha \in A$ such that $y_{\beta_1} = x_{\alpha_1}$.)
The equivalence between these definitions is as follows:
So from the perspective of definition (3), there are enough (1)-subnets and (2)-subnets, up to equivalence.
A textbook account is in
Lecture notes include